RD Sharma Solutions for CBSE Class 10 Mathematics chapter 11 - Trigonometric Identities

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Chapter 11 - Trigonometric Identities Excercise Ex. 11.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Prove the following trigonometric identities:

 

Solution 39

L.H.S.,

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Prove that:

 

(sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Prove that:

 

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

begin mathsize 12px style If space 3 space sin space straight theta space plus space 5 space cos space straight theta equals 5 comma space prove space that space 5 space sin space straight theta minus space 3 space cosθ equals plus-or-minus 3. end style

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73


Question 74

Solution 74

Question 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

Question 80

Solution 80

Question 81

Solution 81

Question 82

Solution 82

Question 83

Solution 83

Question 84

Solution 84


Question 85

Solution 85

Question 86

Solution 86

Question 87

Solution 87

Question 88

Solution 88

Question 89

Solution 89

Question 90

Solution 90

Question 91

Solution 91

Question 92

Solution 92

Question 93

Solution 93

Question 94

Solution 94

Question 95

Solution 95

Question 96

If sin θ + 2 cos θ = 1 prove that 2 sin θ - cos θ = 2.

Solution 96

Chapter 11 - Trigonometric Identities Excercise Ex. 11.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

If 2sin2 θ - cos2 θ = 2, then find the value of θ.

Solution 13

Question 14

If  tan θ - 1 = 0, find the value of sin2θ - cos2 θ.

Solution 14

Chapter 11 - Trigonometric Identities Excercise 11.56

Question 1

begin mathsize 11px style table attributes columnalign left end attributes row cell If text  sec end text straight theta text  + tan end text straight theta text  = x then sec end text straight theta end cell row cell left parenthesis straight a right parenthesis text   end text fraction numerator straight x squared plus 1 over denominator straight x end fraction text        end text left parenthesis straight b right parenthesis text   end text fraction numerator straight x squared plus 1 over denominator 2 straight x end fraction text      end text left parenthesis straight c right parenthesis fraction numerator straight x squared minus 1 over denominator 2 straight x end fraction text       end text left parenthesis straight d right parenthesis text   end text fraction numerator straight x squared minus 1 over denominator straight x end fraction end cell end table end style

Solution 1

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text    tan end text to the power of text 2 end text end exponent straight theta text  = sec end text to the power of text 2 end text end exponent straight theta minus 1 end cell row cell text    end text rightwards double arrow text  sec end text to the power of text 2 end text end exponent straight theta minus tan squared straight theta equals 1 end cell row cell text    end text rightwards double arrow left parenthesis secθ minus tanθ right parenthesis text   end text left parenthesis secθ plus tanθ right parenthesis equals 1 end cell row cell text    end text rightwards double arrow secθ minus tanθ equals fraction numerator 1 over denominator secθ plus tanθ end fraction text      -- end text box enclose 1 end cell row cell Given text  sec end text straight theta text  + tan end text straight theta text  = x    --- end text box enclose 2 end cell row cell so comma text   sec end text straight theta text  - tan end text straight theta text  =  end text 1 over straight x text     --- end text box enclose 3 end cell row cell Apply text   end text box enclose 2 text   +   end text box enclose 3 end cell row cell rightwards double arrow 2 secθ equals straight x plus 1 over straight x end cell row cell rightwards double arrow 2 secθ equals fraction numerator straight x squared plus 1 over denominator straight x end fraction end cell row cell rightwards double arrow box enclose secθ equals fraction numerator straight x squared plus 1 over denominator 2 straight x end fraction end enclose end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 11px style table attributes columnalign left end attributes row cell If text   end text secθ text   end text plus text   end text tanθ equals text   end text straight x comma text   end text then text   end text tanθ end cell row cell left parenthesis straight a right parenthesis fraction numerator straight x squared plus 1 over denominator straight x end fraction text       end text left parenthesis straight b right parenthesis fraction numerator straight x squared minus 1 over denominator straight x end fraction text      end text left parenthesis straight c right parenthesis text   end text fraction numerator straight x squared plus 1 over denominator 2 straight x end fraction text     end text left parenthesis straight d right parenthesis fraction numerator straight x squared minus 1 over denominator 2 straight x end fraction end cell end table end style

Solution 2

begin mathsize 11px style table attributes columnalign left end attributes row cell Given space secθ  +  tanθ equals space straight x space space space space space minus negative negative box enclose 1 end cell row cell text ⇒ end text fraction numerator 1 over denominator space secθ space plus space tanθ end fraction equals space 1 over straight x end cell row cell text ⇒ end text fraction numerator space secθ space minus space tanθ over denominator space secθ squared space minus space tanθ squared end fraction equals space 1 over straight x end cell row cell text ⇒ end text space secθ space minus space tanθ equals space 1 over straight x.... left parenthesis Since space space secθ squared space equals 1 plus space tanθ squared right parenthesis end cell end table
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative box enclose 2
table attributes columnalign left end attributes row cell Apply straight space box enclose 1    -   box enclose 2 end cell row cell rightwards double arrow space space 2 tanθ space equals space straight x minus 1 over straight x end cell row cell rightwards double arrow straight space box enclose tanθ equals fraction numerator straight x squared minus 1 over denominator 2 straight x end fraction end enclose end cell end table
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 3

begin mathsize 11px style table attributes columnalign left end attributes row cell square root of fraction numerator 1 plus sinθ over denominator 1 minus sinθ end fraction end root text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  sec end text straight theta plus tanθ text        end text left parenthesis straight b right parenthesis text  sec end text straight theta minus tanθ text   end text end cell row cell left parenthesis straight c right parenthesis text  sec end text to the power of text 2 end text end exponent straight theta plus tan squared straight theta text      end text left parenthesis straight d right parenthesis text  sec end text to the power of text 2 end text end exponent straight theta minus tan squared straight theta end cell end table end style

Solution 3

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text      end text sin squared straight theta plus cos squared equals 1 end cell row cell rightwards double arrow text   end text box enclose cos squared equals 1 minus sin squared straight theta end enclose text     --- end text box enclose 1 end cell row cell Given text   end text square root of fraction numerator 1 plus sinθ over denominator 1 minus sinθ end fraction end root end cell row cell text     end text rightwards double arrow text   end text square root of fraction numerator left parenthesis 1 plus sinθ right parenthesis text   end text left parenthesis 1 plus sinθ right parenthesis over denominator left parenthesis 1 minus sinθ right parenthesis text   end text left parenthesis 1 plus sinθ right parenthesis end fraction end root end cell row cell text    end text rightwards double arrow text   end text square root of fraction numerator left parenthesis 1 plus sinθ right parenthesis squared over denominator 1 minus sin squared straight theta end fraction end root end cell row cell from text   end text box enclose 1 end cell row cell text    end text rightwards double arrow text    end text square root of fraction numerator left parenthesis 1 plus sinθ right parenthesis squared over denominator cos squared straight theta end fraction end root text       end text end cell row cell text    end text rightwards double arrow text   end text fraction numerator 1 plus sinθ over denominator cosθ end fraction end cell row cell text    end text rightwards double arrow text   end text 1 over cosθ text   end text plus text   end text sinθ over cosθ end cell row cell text    end text rightwards double arrow text  sec end text straight theta text  + tan end text straight theta end cell end table
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

begin mathsize 11px style table attributes columnalign left end attributes row cell The text  value of  end text square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root text  is  end text end cell row cell left parenthesis straight a right parenthesis text  cos end text straight theta plus cosecθ text        end text left parenthesis straight b right parenthesis text   end text cosecθ plus cotθ text   end text end cell row cell left parenthesis straight c right parenthesis text  cosec end text to the power of text 2 end text end exponent straight theta plus cot squared straight theta text      end text left parenthesis straight d right parenthesis text   end text left parenthesis cotθ plus cosecθ right parenthesis squared end cell end table end style

Solution 4

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text      cos end text squared straight theta plus sin squared equals 1 end cell row cell rightwards double arrow text   end text box enclose sin squared straight theta equals 1 minus cso squared straight theta end enclose text     --- end text box enclose 1 end cell row cell Given text   end text square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root end cell row cell text     end text equals text   end text square root of fraction numerator left parenthesis 1 plus cosθ right parenthesis text   end text left parenthesis 1 plus cosθ right parenthesis over denominator left parenthesis 1 minus cosθ right parenthesis text   end text left parenthesis 1 plus cosθ right parenthesis end fraction end root end cell row cell text     end text equals text   end text square root of fraction numerator left parenthesis 1 plus cosθ right parenthesis squared over denominator 1 minus cos squared straight theta end fraction end root end cell row cell text     end text equals text    end text square root of fraction numerator left parenthesis 1 plus cosθ right parenthesis squared over denominator sin squared straight theta end fraction end root text      from equation   end text box enclose 1 end cell row cell text     end text equals text   end text fraction numerator 1 plus cosθ over denominator sinθ end fraction end cell row cell text     end text equals text   end text 1 over sinθ text   end text plus text   end text cosθ over sinθ end cell row cell text     end text equals text  cosec end text straight theta text  + cot end text straight theta end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

sec4 - sec2A  is equal to

(a)  tan2A -  tan4A     (b) tan4A -  tan2

(c) tan4A + tan2A      (d) tan2A -  tan4A  

Solution 5

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text       end text 1 text  + tan end text to the power of text 2 end text end exponent straight A text  = sec end text to the power of text 2 end text end exponent straight A text      --- end text box enclose 1 end cell row cell rightwards double arrow text  sec end text to the power of text 2 end text end exponent straight A minus 1 equals tan squared straight A text     --- end text box enclose 2 end cell row cell Given colon text  sec end text to the power of text 4 end text end exponent straight A minus sec squared text A end text end cell row cell text         end text equals text  sec end text to the power of text 2 end text end exponent straight A left parenthesis sec squared text A end text minus 1 right parenthesis end cell row cell text from   end text box enclose 1 text   &  end text box enclose 2 end cell row cell text        end text rightwards double arrow text   end text left parenthesis 1 plus tan squared straight A right parenthesis tan squared straight A end cell row cell text        end text equals text   tan end text to the power of text 2 end text end exponent straight A plus text tan end text to the power of 4 straight A end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

cos4A-sin4A is equal  to

(a)  2cos2A+1    (b) 2cos2A-1   

(c)  2sin2A-1     (d) 2sin2A+1  

Solution 6

begin mathsize 11px style We space space know
table attributes columnalign left end attributes row cell text           end text sin squared straight A plus cos squared straight A equals 1 text     --- end text box enclose 1 end cell row cell Given text   cos end text to the power of text 4 end text end exponent straight A minus sin to the power of 4 straight A end cell row cell text       end text equals text   end text left parenthesis cos squared straight A right parenthesis squared minus left parenthesis sin squared straight A right parenthesis squared text                           end text end cell row cell text       end text equals text   end text left parenthesis cos squared straight A plus sin squared straight A right parenthesis text   end text left parenthesis cos squared straight A minus sin squared straight A right parenthesis end cell row cell from text    end text box enclose 1 end cell row cell text       end text rightwards double arrow text   cos end text to the power of text 2 end text end exponent straight A minus sin squared straight A text                    end text left curly bracket table attributes columnalign left end attributes row cell straight a squared minus straight b squared end cell row cell equals left parenthesis straight a minus straight b left parenthesis straight a plus straight b right parenthesis right parenthesis end cell end table right curly bracket end cell row cell text       end text equals text   cos end text to the power of text 2 end text end exponent straight A minus left parenthesis text 1- cos end text to the power of text 2 end text end exponent straight A right parenthesis end cell row cell text       end text equals text   cos end text to the power of text 2 end text end exponent straight A minus text  1 +cos end text to the power of text 2 end text end exponent straight A end cell row cell text       end text equals text   2cos end text to the power of text 2 end text end exponent straight A minus 1 text     end text end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 11 - Trigonometric Identities Excercise 11.57

Question 1

begin mathsize 11px style table attributes columnalign left end attributes row cell fraction numerator sinθ over denominator 1 plus cosθ end fraction text   is equal to end text end cell row cell left parenthesis straight a right parenthesis text   end text fraction numerator 1 plus cosθ over denominator sinθ end fraction text     end text left parenthesis straight b right parenthesis text   end text fraction numerator 1 minus cosθ over denominator cosθ end fraction text     end text left parenthesis straight c right parenthesis text   end text fraction numerator 1 minus cosθ over denominator sinθ end fraction text     end text left parenthesis straight d right parenthesis text   end text fraction numerator 1 minus sinθ over denominator cosθ end fraction end cell end table end style

Solution 1

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text             sin end text to the power of text 2 end text end exponent straight theta plus cos squared straight theta equals 1 end cell row cell text        end text rightwards double arrow text  1-  end text cos squared straight theta equals text  sin end text to the power of text 2 end text end exponent straight theta text      --- end text box enclose 1 text   end text end cell row cell text Given :  end text fraction numerator sinθ over denominator 1 plus cosθ end fraction end cell row cell text       end text equals text    end text fraction numerator sinθ left parenthesis 1 minus cosθ right parenthesis over denominator left parenthesis 1 plus cosθ right parenthesis text   end text left parenthesis 1 minus cosθ right parenthesis end fraction end cell row cell text       end text equals text    end text fraction numerator sinθ left parenthesis 1 minus cosθ right parenthesis over denominator text  1-cos end text to the power of text 2 end text end exponent straight theta end fraction end cell row cell from text   end text box enclose 1 end cell row cell text       end text rightwards double arrow text   end text fraction numerator sinθ left parenthesis 1 minus cosθ right parenthesis over denominator text  sin end text to the power of text 2 end text end exponent straight theta end fraction end cell row cell text       end text equals text   end text fraction numerator 1 minus cosθ over denominator sinθ end fraction end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 11px style table attributes columnalign left end attributes row cell fraction numerator sinθ over denominator 1 minus cotθ end fraction plus fraction numerator cosθ over denominator 1 minus tanθ end fraction text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  0      end text left parenthesis straight b right parenthesis text  1       end text left parenthesis straight c right parenthesis text  sin end text straight theta text  + cos end text straight theta text     end text left parenthesis straight d right parenthesis text  sin end text straight theta text  - cos end text straight theta end cell end table end style

Solution 2

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text            tan end text straight theta text  =  end text sinθ over cosθ text      --- end text box enclose 1 end cell row cell text            cot end text straight theta text  =  end text cosθ over sinθ text      --- end text box enclose 2 end cell row cell Given colon text   end text fraction numerator sinθ over denominator 1 minus cosθ end fraction plus fraction numerator cosθ over denominator 1 minus tanθ end fraction end cell row cell from text    end text box enclose 1 text  &  end text box enclose 2 end cell row cell text     end text rightwards double arrow text    end text fraction numerator sinθ over denominator 1 minus cosθ over sinθ end fraction text   end text plus text   end text fraction numerator cosθ over denominator 1 minus sinθ over cosθ end fraction end cell row cell text    end text equals text    end text fraction numerator sin squared straight theta over denominator sinθ minus cosθ end fraction text   end text plus text    end text fraction numerator cos squared straight theta over denominator cosθ minus sinθ end fraction end cell row cell text    end text equals text   end text fraction numerator sin squared straight theta minus text   end text cos squared straight theta text   end text over denominator sinθ minus text   end text cosθ text   end text end fraction end cell row cell text    end text equals text   end text fraction numerator left parenthesis sinθ minus up diagonal strike straight c osθ right parenthesis text   end text left parenthesis sinθ plus cosθ right parenthesis over denominator left parenthesis sinθ minus up diagonal strike straight c osθ right parenthesis end fraction end cell row cell text    end text equals text   end text left parenthesis sinθ plus text cos end text straight theta right parenthesis end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

The value of (1+cotθ – cosecθ )  (1+tanθ + secθ) is

 (a) 1     (b) 2      (c) 4      (d) 0

Solution 3

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell cotθ equals text   end text cosθ over sinθ text     --- end text box enclose 1 text        end text cosecθ equals text   end text 1 over sinθ text     --- end text box enclose 3 end cell row cell tanθ equals text   end text sinθ over cosθ text     --- end text box enclose 2 text        end text secθ equals text   end text 1 over cosθ text     --- end text box enclose 4 text    end text end cell row cell text Given :  end text left parenthesis 1 plus cotθ minus text  cosec end text straight theta right parenthesis text   end text left parenthesis 1 plus tanθ plus secθ right parenthesis text   end text end cell row cell from text    end text box enclose 1 text ,    end text box enclose 2 text ,    end text box enclose 3 text ,    end text box enclose 4 end cell row cell rightwards double arrow left parenthesis 1 plus cosθ over sinθ minus 1 over sinθ right parenthesis text   end text left parenthesis 1 plus sinθ over cosθ plus 1 over cosθ right parenthesis end cell row cell rightwards double arrow left parenthesis fraction numerator sinθ plus cosθ minus 1 over denominator sinθ end fraction right parenthesis text   end text left parenthesis fraction numerator sinθ plus cosθ plus 1 over denominator cosθ end fraction right parenthesis end cell row cell rightwards double arrow fraction numerator left parenthesis sinθ plus cosθ right parenthesis squared minus 1 over denominator sinθcosθ end fraction text                      end text left curly bracket left parenthesis straight a plus straight b right parenthesis left parenthesis straight a minus straight b right parenthesis equals straight a squared minus straight b squared right curly bracket end cell row cell rightwards double arrow fraction numerator sin squared straight theta plus cos squared plus 2 sinθcosθ minus 1 over denominator sinθcosθ end fraction equals 2 text    end text left curly bracket sin squared straight theta plus cos squared straight theta equals 1 right curly bracket end cell row cell rightwards double arrow fraction numerator 1 plus 2 sinθcosθ minus 1 over denominator sinθcosθ end fraction equals 2 end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis.
end style

Question 4

begin mathsize 11px style table attributes columnalign left end attributes row cell fraction numerator tanθ over denominator secθ minus 1 end fraction plus fraction numerator tanθ over denominator secθ plus 1 end fraction text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  2 tan end text straight theta text    end text left parenthesis straight b right parenthesis text  2 sec end text straight theta text    end text left parenthesis straight c right parenthesis text  2 cosec end text straight theta text    end text left parenthesis straight d right parenthesis text  2 tan end text straight theta text  sec end text straight theta end cell end table end style

Solution 4

begin mathsize 11px style table attributes columnalign left end attributes row cell text      end text fraction numerator tanθ over denominator secθ minus 1 end fraction plus fraction numerator tanθ over denominator secθ plus 1 end fraction end cell row cell rightwards double arrow text   end text fraction numerator tanθ left parenthesis secθ plus 1 right parenthesis plus tanθ left parenthesis secθ minus 1 right parenthesis over denominator left parenthesis secθ minus 1 right parenthesis left parenthesis secθ plus 1 right parenthesis end fraction text   end text end cell row cell rightwards double arrow text   end text fraction numerator tanθ text   end text secθ plus tanθ plus tanθ text   end text secθ minus tanθ over denominator sec squared straight theta minus 1 end fraction end cell row cell rightwards double arrow text   end text fraction numerator 2 tanθsecθ over denominator tan squared straight theta end fraction text                     end text left curly bracket sec squared straight theta equals 1 plus tan squared straight theta right curly bracket end cell row cell rightwards double arrow text   end text fraction numerator 2 secθ over denominator tanθ end fraction end cell row cell rightwards double arrow text   end text fraction numerator 2 left parenthesis 1 over cosθ right parenthesis over denominator sinθ over cosθ end fraction end cell row cell rightwards double arrow text   end text 2 over sinθ equals 2 cosecθ end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 5

(cosecθ - sinθ) (secθ - cosθ) (tanθ + cotθ) is equal

(a) 0          (b)   1       (c)-1        (d) None of thee

Solution 5

begin mathsize 11px style table attributes columnalign left end attributes row cell We text  know  end text box enclose sin squared straight theta plus cos squared straight theta plus 1 end enclose text          --- end text box enclose 1 end cell row cell text   end text left parenthesis cosecθ minus sinθ right parenthesis left parenthesis secθ minus cosθ right parenthesis left parenthesis tanθ plus cotθ right parenthesis end cell row cell equals text   end text left parenthesis 1 over sinθ minus sinθ right parenthesis left parenthesis 1 over cosθ minus cosθ right parenthesis left parenthesis sinθ over cosθ plus cosθ over sinθ right parenthesis end cell row cell equals text   end text fraction numerator left parenthesis 1 minus sin squared straight theta right parenthesis over denominator sinθ end fraction cross times fraction numerator left parenthesis 1 minus cos squared straight theta right parenthesis over denominator cosθ end fraction cross times fraction numerator left parenthesis sinθ plus cos squared straight theta right parenthesis over denominator sinθcosθ end fraction end cell row cell from text   end text box enclose 1 end cell row cell rightwards double arrow text   end text fraction numerator cos squared straight theta over denominator sinθ end fraction cross times fraction numerator sin squared straight theta over denominator cosθ end fraction cross times 1 over sinθcosθ end cell row cell equals text  1 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style table attributes columnalign left end attributes row cell If space text   end text straight x equals acosθ space space and text    end text space straight y equals text   end text bsinθ comma text   end text than text   end text straight b squared straight x squared plus straight a squared straight y squared equals end cell row cell left parenthesis straight a right parenthesis straight a squared straight b squared space space space space space left parenthesis straight b right parenthesis ab space space space space space space left parenthesis straight c right parenthesis text   end text straight a to the power of 4 straight b to the power of 4 space space space left parenthesis straight d right parenthesis text   end text straight a squared plus straight b squared end cell end table end style

Solution 6

begin mathsize 11px style Given space straight x equals acosθ space space space space and space space space straight y equals bsinθ
table attributes columnalign left end attributes row cell space space space space straight x squared equals straight a squared cos squared straight theta space space space space and space space space straight y to the power of straight 2 equals straight b squared sin squared straight theta end cell row cell rightwards double arrow space straight b to the power of straight 2 straight x squared equals straight a squared straight b to the power of straight 2 cos squared straight theta space space space and space space straight a squared straight y to the power of straight 2 equals straight a squared straight b to the power of straight 2 sin squared straight theta end cell row cell rightwards double arrow space straight b to the power of straight 2 straight x squared plus straight space straight a squared straight y to the power of straight 2 equals straight space straight a squared straight b to the power of straight 2 straight space left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis end cell row cell rightwards double arrow space straight b to the power of straight 2 straight x squared plus straight space straight a squared straight y to the power of straight 2 equals straight space straight a squared straight b to the power of straight 2 left parenthesis 1 right parenthesis end cell row cell space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight a squared straight b to the power of straight 2 end cell end table
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 7

If  x =a secθ  and  y=b tanθ  then  b2x2a2y2=

(a) ab      (b) a- b2       (c) a+ b2    (d) a2b2

Solution 7

begin mathsize 11px style table attributes columnalign left end attributes row cell We straight space know space space    end cell row cell         sec squared straight theta straight space – straight space tan squared straight theta equals 1 space space     --- box enclose 1 end cell row cell straight space rightwards double arrow space straight b to the power of straight 2 straight x squared straight space minus space straight a to the power of straight 2 straight y squared end cell row cell straight space equals space straight b to the power of straight 2 left parenthesis straight a straight space secθ right parenthesis squared minus space straight a to the power of straight 2 left parenthesis straight b straight space tanθ right parenthesis squared end cell row cell straight space equals space straight a to the power of straight 2 straight b to the power of straight 2 straight space sec squared straight theta minus straight a to the power of straight 2 straight b to the power of straight 2 straight space tan squared straight theta end cell row cell straight space equals space straight a to the power of straight 2 straight b to the power of straight 2 straight space left parenthesis sec squared straight theta straight space – straight space tan squared straight theta right parenthesis end cell row cell straight space space from space box enclose 1 end cell row cell straight space rightwards double arrow space straight a to the power of straight 2 straight b to the power of straight 2 straight space left parenthesis sec squared straight theta straight space – straight space tan squared straight theta right parenthesis equals space straight a to the power of straight 2 straight b to the power of straight 2 straight space end cell end table
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 8

begin mathsize 11px style table attributes columnalign left end attributes row cell fraction numerator cotθ over denominator cotθ minus cot 30 end fraction text   end text plus text   end text fraction numerator tanθ over denominator tanθ minus tan 30 end fraction text      is equal to end text end cell row cell left parenthesis straight a right parenthesis text  0       end text left parenthesis straight b right parenthesis 1 text        end text left parenthesis straight c right parenthesis minus 1 text        end text left parenthesis straight d right parenthesis text  2 end text end cell end table end style

Solution 8

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text         cot 3 end text straight theta text  =  end text fraction numerator 3 cotθ minus cot cubed straight theta over denominator 1 minus 3 cot squared straight theta end fraction text    ---- end text box enclose 1 end cell row cell text         tan 3 end text straight theta text  =  end text fraction numerator 3 tanθ minus tan cubed straight theta over denominator 1 minus 3 tan squared straight theta end fraction text    ---- end text box enclose 2 end cell row cell rightwards double arrow text   end text fraction numerator cotθ over denominator cotθ minus cot 3 straight theta end fraction text   end text plus text   end text fraction numerator tanθ over denominator tanθ minus tan 3 straight theta end fraction end cell row cell from text     end text box enclose 1 text    &   end text box enclose 2 text   end text end cell row cell rightwards double arrow text   end text fraction numerator cotθ over denominator cotθ minus left parenthesis fraction numerator 3 cotθ minus cot cubed straight theta over denominator 1 minus 3 cot squared straight theta end fraction right parenthesis end fraction text   end text plus text   end text fraction numerator tanθ over denominator tanθ minus left parenthesis fraction numerator 3 tanθ minus tan cubed straight theta over denominator 1 minus 3 tan squared straight theta end fraction right parenthesis end fraction end cell row cell equals text   end text fraction numerator cotθ left parenthesis 1 minus 3 cot squared straight theta right parenthesis over denominator negative 2 cotθ minus 2 cot cubed straight theta end fraction text  +  end text fraction numerator tanθ left parenthesis 1 minus 3 tan squared straight theta right parenthesis over denominator negative 2 tanθ minus 2 tan cubed straight theta end fraction end cell row cell equals text  - end text fraction numerator left parenthesis 1 minus 3 tan squared straight theta right parenthesis over denominator 2 left parenthesis 1 plus tan squared straight theta right parenthesis end fraction text   end text minus text   end text fraction numerator left parenthesis 1 minus 3 cot squared straight theta right parenthesis over denominator 2 left parenthesis 1 plus cot squared straight theta right parenthesis end fraction end cell row cell equals text  - end text 1 half left curly bracket fraction numerator 1 minus 3 tan squared straight theta over denominator sec squared straight theta end fraction text  +  end text fraction numerator 1 minus 3 cot squared straight theta over denominator cosec squared straight theta end fraction text   end text right curly bracket end cell row cell equals text  - end text 1 half left curly bracket fraction numerator 1 minus fraction numerator 3 sin squared straight theta over denominator cos squared straight theta end fraction over denominator fraction numerator 1 over denominator cos squared straight theta end fraction end fraction text  +  end text fraction numerator 1 minus fraction numerator 3 cos squared straight theta over denominator sin squared straight theta end fraction over denominator fraction numerator 1 over denominator cos squared straight theta end fraction end fraction text   end text right curly bracket end cell row cell equals text  - end text 1 half left curly bracket cosθ minus 3 sin squared straight theta text  + sin end text to the power of text 2 end text end exponent straight theta minus 3 cos squared straight theta right curly bracket end cell row cell equals text  - end text 1 half left curly bracket negative 2 cos squared straight theta text  - 2sin end text to the power of text 2 end text end exponent straight theta right curly bracket end cell row cell equals text  sin end text to the power of text 2 end text end exponent straight theta text  + cos end text to the power of text 2 end text end exponent straight theta end cell row cell equals text  1 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 9

begin mathsize 11px style table attributes columnalign left end attributes row cell 2 left parenthesis sin to the power of 6 straight theta plus cos to the power of 6 straight theta right parenthesis minus 3 left parenthesis sin to the power of 4 straight theta plus cos to the power of 4 straight theta right parenthesis text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  0     end text left parenthesis straight b right parenthesis text  1      end text left parenthesis straight c right parenthesis text  -1      end text left parenthesis straight d right parenthesis text  None of these end text end cell end table end style

Solution 9

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Question 10

If acosθ + bsinθ = 4 and asinθ-bcosθ = 3, then a2+b2=

(a) 7    (b) 12      (c) 25      (d) None of these

Solution 10

begin mathsize 11px style table attributes columnalign left end attributes row cell acosθ plus bsinθ equals 4 end cell row cell On text  squaring both sides end text end cell row cell rightwards double arrow text  a end text to the power of text 2 end text end exponent cos squared straight theta plus straight b squared sin squared straight theta plus 2 absinθcosθ equals 16 text      --- end text box enclose 1 end cell row cell text      asin end text straight theta text  - bcos end text straight theta equals 3 end cell row cell text On squaring both sides end text end cell row cell rightwards double arrow text  a end text to the power of text 2 end text end exponent sin squared straight theta plus straight b squared cos squared straight theta minus 2 absincosθ equals 9 text       --- end text box enclose 2 end cell row cell Adding text    end text box enclose 1 text   &  end text box enclose 2 text   end text end cell row cell rightwards double arrow text  a end text to the power of text 2 end text end exponent left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis plus straight b squared left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis equals 25 end cell row cell rightwards double arrow text  a end text to the power of text 2 end text end exponent plus straight b squared equals 25 end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 11

begin mathsize 11px style table attributes columnalign left end attributes row cell If text   end text acotθ plus bcosecθ equals straight p text   and bcot+ end text straight theta plus acosecθ equals straight q comma text    end text end cell row cell text then p end text to the power of text 2 end text end exponent text-end text straight q squared equals end cell row cell left parenthesis straight a right parenthesis text  a end text to the power of text 2 end text end exponent minus straight b to the power of text 2 end text end exponent text      end text left parenthesis straight b right parenthesis text  b end text to the power of text 2 end text end exponent minus text a end text to the power of text 2 end text end exponent text       end text left parenthesis straight c right parenthesis text  a end text to the power of text 2 end text end exponent plus straight b squared text        end text left parenthesis straight d right parenthesis text  b-a end text end cell end table end style

Solution 11

begin mathsize 11px style table attributes columnalign left end attributes row cell Given straight space acotθ plus bcosecθ equals straight p end cell row cell Squaring straight space both space sides end cell row cell rightwards double arrow space straight a to the power of straight 2 cot squared straight theta plus straight b squared cosec squared straight theta plus 2 abcotθcosecθ equals straight p squared     --- box enclose 1 end cell row cell Given space bcotθ  +  acosecθ equals straight q end cell row cell Squaring space both space sides end cell row cell rightwards double arrow space straight b to the power of straight 2 cot squared straight theta plus straight a squared cosec squared straight theta plus 2 abcotθcoseθ equals straight q squared     --- box enclose 2 end cell row cell Subtracting space space box enclose 2 space from space box enclose 1 comma end cell row cell rightwards double arrow space straight a to the power of straight 2 left parenthesis cot squared straight theta minus cosec squared straight theta right parenthesis plus straight b squared left parenthesis cosec squared straight theta minus cot squared straight theta right parenthesis equals straight p squared minus straight q squared   --- box enclose 3 end cell row cell We space know space space cosec squared straight theta equals 1 plus straight space cot squared straight theta end cell row cell space rightwards double arrow cosec squared straight theta minus cot squared straight theta equals 1        --- box enclose 4    end cell row cell from space box enclose 3   &   box enclose 4 end cell row cell rightwards double arrow negative straight a to the power of straight 2 plus straight b to the power of straight 2 equals straight p squared minus straight q squared end cell row cell rightwards double arrow space space    straight p squared minus straight q squared equals straight b to the power of straight 2 minus straight a to the power of straight 2 end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 12

begin mathsize 11px style table attributes columnalign left end attributes row cell The text   end text value text   end text of text   end text sin squared 29 to the power of 0 plus sin squared 61 to the power of 0 text  is end text end cell row cell left parenthesis straight a right parenthesis text  1        end text left parenthesis straight b right parenthesis text  0        end text left parenthesis straight c right parenthesis text  2sin end text to the power of text 2 end text end exponent 29 to the power of 0 text      end text left parenthesis straight d right parenthesis text  2cos end text to the power of text 2 end text end exponent 61 to the power of 0 end cell end table end style

Solution 12

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text      end text sin left parenthesis 90 minus straight theta right parenthesis equals cos left parenthesis straight theta right parenthesis end cell row cell rightwards double arrow text  for  end text straight theta text  = 29 end text to the power of text 0 end text end exponent end cell row cell rightwards double arrow text  sin end text left parenthesis 61 to the power of 0 right parenthesis equals cos 29 to the power of 0 text       --- end text box enclose 1 end cell row cell text      end text also text   sin end text to the power of text 2 end text end exponent straight theta plus cos squared straight theta equals 1 text    --- end text box enclose 2 end cell row cell rightwards double arrow text  sin end text to the power of text 2 end text end exponent text 29 end text to the power of text 0 end text end exponent plus sin squared 61 to the power of 0 end cell row cell text      end text from text      end text box enclose 1 end cell row cell rightwards double arrow text  sin end text to the power of text 2 end text end exponent text 29 end text to the power of text 0 end text end exponent plus cos to the power of text 2 end text end exponent text 29 end text to the power of text 0 end text end exponent end cell row cell rightwards double arrow text  1 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 13

begin mathsize 11px style table attributes columnalign left end attributes row cell If text   end text straight x text   end text equals text   end text rsinθcosϕ comma text  y=  end text rsinθsinϕ text   and z= rcos end text straight theta text , then end text end cell row cell left parenthesis straight a right parenthesis text  x end text to the power of text 2 end text end exponent plus straight y squared plus straight z squared equals straight r squared text         end text left parenthesis straight b right parenthesis text  x end text to the power of text 2 end text end exponent plus straight y squared minus straight z squared equals straight r squared end cell row cell left parenthesis straight c right parenthesis text  x end text to the power of text 2 end text end exponent minus straight y squared plus straight z squared equals straight r squared text         end text left parenthesis straight d right parenthesis text  z end text to the power of text 2 end text end exponent plus straight y squared plus straight x squared equals straight r squared end cell end table end style

Solution 13

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals straight space rsinθcosϕ end cell row cell straight y equals straight space rsinθsinϕ end cell row cell straight z equals space rcosθ end cell row cell We space know space space space sin to the power of straight 2 straight A plus cos squared straight A equals 1   --- box enclose 1 end cell row cell rightwards double arrow space straight x to the power of straight 2 plus straight y to the power of straight 2 end cell row cell rightwards double arrow straight space straight r squared sin squared θcos squared straight ϕ plus straight r squared sin squared θsin squared straight ϕ end cell row cell rightwards double arrow straight space straight r squared sin squared straight theta left parenthesis cos squared straight ϕ plus sin squared straight ϕ right parenthesis end cell row cell rightwards double arrow straight space straight r squared sin squared straight theta          --- box enclose 1 end cell row cell rightwards double arrow straight space straight r squared sin squared straight theta straight space equals space straight x to the power of straight 2 plus straight y to the power of straight 2 end cell row cell Add space straight z to the power of straight 2 space both space side end cell row cell rightwards double arrow space straight x to the power of straight 2 plus straight y to the power of straight 2 plus straight z to the power of straight 2 equals straight r squared sin squared straight theta plus straight r squared cos squared straight theta end cell row cell                space space space space space     =  straight r squared left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis               end cell row cell rightwards double arrow space straight x to the power of straight 2 plus straight y to the power of straight 2 plus straight z to the power of straight 2 equals straight r squared end cell end table
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 11 - Trigonometric Identities Excercise 11.58

Question 1

begin mathsize 11px style table attributes columnalign left end attributes row cell If text   end text sinθ plus sin squared straight theta equals 1 text   then  cos end text to the power of text 2 end text end exponent straight theta plus cos to the power of 4 straight theta equals end cell row cell left parenthesis straight a right parenthesis text  -1     end text left parenthesis straight b right parenthesis text  1      end text left parenthesis straight c right parenthesis text  0       end text left parenthesis straight d right parenthesis text  None of these end text end cell end table end style

Solution 1

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text    end text sin squared straight theta plus cos squared straight theta equals 1 end cell row cell rightwards double arrow cos squared straight theta equals 1 minus sin squared straight theta text             --- end text box enclose 1 end cell row cell Given text     s end text inθ plus sin squared straight theta text  =1    --- end text box enclose 2 text   end text end cell row cell text              end text sinθ equals text  1- end text sin squared straight theta text   end text end cell row cell text              from  end text box enclose 1 end cell row cell text               end text box enclose sinθ equals cos squared straight theta end enclose text       --- end text box enclose 3 end cell row cell To text  find    end text cos squared straight theta plus cos to the power of 4 straight theta end cell row cell text           end text rightwards double arrow cos squared straight theta left parenthesis 1 plus cos squared straight theta right parenthesis end cell row cell text           from   end text box enclose 3 end cell row cell text           end text rightwards double arrow sinθ left parenthesis 1 plus sinθ right parenthesis end cell row cell text           end text rightwards double arrow sinθ plus sin squared straight theta end cell row cell text           from  end text box enclose 2 end cell row cell text            end text rightwards double arrow text  1 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 11px style table attributes columnalign left end attributes row cell If text   end text acosθ plus bsinθ equals straight m text   and asin end text straight theta text -bcos end text straight theta text  = n, then a end text to the power of text 2 end text end exponent plus straight b squared equals end cell row cell left parenthesis straight a right parenthesis text  m end text to the power of text 2 end text end exponent minus straight n squared text      end text left parenthesis straight b right parenthesis text  m end text to the power of text 2 end text end exponent straight n squared text     end text left parenthesis straight c right parenthesis text   end text straight n squared text  - m end text to the power of text 2 end text end exponent text     end text left parenthesis straight d right parenthesis text  m end text to the power of text 2 end text end exponent plus straight n squared text   end text end cell end table end style

Solution 2

begin mathsize 11px style table attributes columnalign left end attributes row cell acosθ plus bsinθ equals straight m end cell row cell On text  squaring end text end cell row cell text a end text to the power of text 2 end text end exponent cos squared straight theta plus straight b squared sin squared straight theta plus 2 absinθcosθ equals straight m squared text     --- end text box enclose 1 end cell row cell asinθ minus bcosθ equals straight n end cell row cell On text  squaring end text end cell row cell text a end text to the power of text 2 end text end exponent sin squared straight theta plus straight b squared cos squared straight theta minus 2 absinθcosθ equals straight n squared text       --- end text box enclose 2 end cell row cell On text   end text adding text   end text box enclose 1 text  &  end text box enclose 2 text   and knowing sin end text to the power of text 2 end text end exponent straight theta plus cos squared straight theta equals 1 end cell row cell rightwards double arrow text a end text to the power of text 2 end text end exponent left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis plus straight b squared left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis equals straight m squared plus straight n squared end cell row cell rightwards double arrow text a end text to the power of text 2 end text end exponent plus straight b to the power of text 2 end text end exponent equals straight m to the power of text 2 end text end exponent plus straight n to the power of text 2 end text end exponent end cell end table
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 3

begin mathsize 11px style table attributes columnalign left end attributes row cell If text  cosA + cos end text to the power of text 2 end text end exponent straight A equals 1 comma text   then  sin end text to the power of text 2 end text end exponent straight A plus sin to the power of 4 straight A equals end cell row cell left parenthesis straight a right parenthesis text  -1       end text left parenthesis straight b right parenthesis text  0        end text left parenthesis straight c right parenthesis text  1       end text left parenthesis straight d right parenthesis text  None of these end text end cell end table end style

Solution 3

begin mathsize 11px style table attributes columnalign left end attributes row cell text            end text sin squared straight A plus cos squared straight A equals 1 end cell row cell text            end text sin squared straight A equals 1 minus cos squared straight A text            --- end text box enclose 1 end cell row cell given text    cosA + cos end text to the power of text 2 end text end exponent straight A equals 1 text             --- end text box enclose 2 end cell row cell text           cosA = 1-cos end text to the power of text 2 end text end exponent straight A end cell row cell text        from  end text box enclose text 1 end text end enclose end cell row cell text           cosA end text equals text sin end text to the power of text 2 end text end exponent text A                   --- end text box enclose 3 end cell row cell To text  find:    s end text in squared straight A plus sin to the power of 4 straight A end cell row cell text             end text rightwards double arrow sin squared straight A left parenthesis 1 plus sin squared straight A right parenthesis end cell row cell text           end text from text   end text box enclose 3 text     end text end cell row cell text              end text rightwards double arrow text  cosA end text left parenthesis 1 plus text cosA end text right parenthesis end cell row cell text              end text rightwards double arrow text  cosA end text plus text cos end text to the power of text 2 end text end exponent text A end text end cell row cell text            end text from text   end text box enclose 2 end cell row cell text               end text rightwards double arrow text  1 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 4

begin mathsize 11px style table attributes columnalign left end attributes row cell If text  x = asec end text θcosϕ comma text  y = bsec end text θsinϕ text   and z = ctan end text straight theta end cell row cell text then   end text straight x squared over straight a squared text   end text plus text   end text straight y squared over straight b squared equals end cell row cell left parenthesis straight a right parenthesis text   end text straight z squared over straight c squared text      end text left parenthesis straight b right parenthesis text  1- end text straight z squared over straight c squared text        end text left parenthesis straight c right parenthesis text   end text straight z squared over straight c squared minus 1 text       end text left parenthesis straight d right parenthesis text  1+ end text straight z squared over straight c squared text   end text end cell end table end style

Solution 4

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell sin squared straight theta plus cos squared plus 1 text          --- end text box enclose 1 end cell row cell 1 plus tan squared straight theta equals sec squared straight theta text      --- end text box enclose 2 end cell row cell rightwards double arrow straight x squared over straight a squared text   end text plus text   end text straight y squared over straight b squared end cell row cell equals fraction numerator straight a squared sec squared θcos squared straight ϕ over denominator straight a squared end fraction text   end text plus text   end text fraction numerator straight b squared sec squared θsin squared straight ϕ over denominator straight b squared end fraction end cell row cell equals text   end text sec squared θcos squared straight ϕ text   end text plus text   end text sec squared θsin squared straight ϕ end cell row cell equals text   end text sec squared straight theta left parenthesis cos squared straight ϕ plus sin squared straight ϕ right parenthesis end cell row cell text  from  end text box enclose 1 text  &  end text box enclose 2 end cell row cell rightwards double arrow text   end text left parenthesis 1 plus tan squared straight theta right parenthesis text          --- end text box enclose 3 end cell row cell given text   z = ctan end text straight theta end cell row cell rightwards double arrow text  tan end text straight theta equals straight z over straight c text    --- end text box enclose 4 end cell row cell rightwards double arrow straight x squared over straight a squared text   end text plus text   end text straight y squared over straight b squared equals 1 plus straight z squared over straight c squared end cell end table
So comma space the space correct space option space is space left parenthesis straight d right parenthesis.
end style

Question 5

begin mathsize 11px style table attributes columnalign left end attributes row cell If text  acos end text straight theta text  - bsin end text straight theta text  = c, then asin end text straight theta text  + bcos end text straight theta equals end cell row cell left parenthesis straight a right parenthesis text   end text plus-or-minus space root index blank of straight a to the power of straight 2 plus straight b to the power of straight 2 plus straight c to the power of straight 2 end root text        end text left parenthesis straight b right parenthesis text   end text plus-or-minus space root index blank of straight a to the power of straight 2 plus straight b to the power of straight 2 minus straight c to the power of straight 2 end root end cell row cell left parenthesis straight c right parenthesis plus-or-minus space root index blank of straight c to the power of straight 2 plus straight a squared minus straight b to the power of straight 2 end root text         end text left parenthesis straight d right parenthesis text  None of these  end text end cell end table end style

Solution 5

begin mathsize 11px style table attributes columnalign left end attributes row cell acosθ minus bsinθ equals straight c end cell row cell rightwards double arrow space space space On space squaring end cell row cell space space space space space space space straight a to the power of straight 2 cos squared straight theta plus straight b squared sin squared straight theta minus 2 abcosθsinθ equals straight c squared space space space minus negative negative box enclose 1 end cell row cell Let space space space asinθ plus bcosθ space equals space straight t end cell row cell space space space space space space space On space squaring end cell row cell space space rightwards double arrow space straight a to the power of straight 2 sin to the power of straight 2 straight theta space plus space straight b to the power of straight 2 cos to the power of straight 2 straight theta space plus space 2 absinθcosθ space equals space straight t to the power of straight 2 space space space minus negative negative box enclose 2 end cell row cell Adding straight space box enclose straight 1 space & space box enclose 2 comma space space we space get end cell row cell rightwards double arrow space straight a to the power of straight 2 left parenthesis sin to the power of straight 2 straight theta plus cos to the power of straight 2 straight theta right parenthesis space plus space straight b to the power of straight 2 left parenthesis sin to the power of straight 2 straight theta plus cos to the power of straight 2 straight theta right parenthesis equals space straight c to the power of straight 2 plus straight t squared space space space minus negative negative box enclose 3 end cell row cell rightwards double arrow space sin to the power of straight 2 straight theta plus cos to the power of straight 2 straight theta equals 1 space space space left parenthesis we space know right parenthesis end cell row cell space space space space space from space box enclose 3 end cell row cell space space space space straight a to the power of straight 2 plus straight b to the power of straight 2 equals straight c to the power of straight 2 plus straight t squared end cell row cell rightwards double arrow space straight t equals plus-or-minus space root index blank of straight a to the power of straight 2 plus straight b to the power of straight 2 minus straight c to the power of straight 2 end root end cell row cell Hence comma end cell row cell asinθ space plus space bcosθ space equals plus-or-minus space root index blank of straight a to the power of straight 2 plus straight b to the power of straight 2 minus straight c to the power of straight 2 end root end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style table attributes columnalign left end attributes row cell 9 sec squared straight A minus 9 tan squared straight A text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  1     end text left parenthesis straight b right parenthesis text  9      end text left parenthesis straight c right parenthesis text  8       end text left parenthesis straight d right parenthesis text  0 end text end cell end table end style

Solution 6

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell text      end text 1 text  + tan end text to the power of text 2 end text end exponent straight a equals sec squared straight A end cell row cell rightwards double arrow sec squared straight A minus tan squared straight A equals 1 text        --- end text box enclose 1 end cell row cell To text  find:  9sec end text to the power of text 2 end text end exponent straight A minus 9 tan squared straight A end cell row cell text            end text rightwards double arrow text  9 end text left parenthesis text sec end text to the power of text 2 end text end exponent straight A minus tan squared straight A right parenthesis end cell row cell text         from  end text box enclose 1 end cell row cell text           end text rightwards double arrow text  9(1) = 9 end text end cell end table
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 7

begin mathsize 11px style table attributes columnalign left end attributes row cell left parenthesis 1 plus tanθ plus secθ right parenthesis left parenthesis 1 plus cotθ minus cosecθ right parenthesis equals end cell row cell left parenthesis straight a right parenthesis text  0     end text left parenthesis straight b right parenthesis text  1      end text left parenthesis straight c right parenthesis text  2        end text left parenthesis straight d right parenthesis text  -1 end text end cell end table end style

Solution 7

begin mathsize 11px style table attributes columnalign left end attributes row cell    left parenthesis 1 plus sinθ over cosθ plus 1 over cosθ right parenthesis left parenthesis 1 plus cosθ over sinθ minus 1 over sinθ right parenthesis end cell row cell equals left parenthesis fraction numerator sinθ plus cosθ plus 1 over denominator cosθ end fraction right parenthesis left parenthesis fraction numerator sinθ plus cosθ minus 1 over denominator sinθ end fraction right parenthesis end cell row cell We space know end cell row cell        left parenthesis straight a plus straight b right parenthesis left parenthesis straight a minus straight b right parenthesis plus straight a squared minus straight b squared          --- box enclose 1 end cell row cell and    sin to the power of straight 2 straight theta plus cos squared plus 1           space   --- box enclose 2 end cell row cell from straight space box enclose 1 comma space we space get end cell row cell             fraction numerator left parenthesis sinθ plus cosθ right parenthesis squared minus 1 over denominator sinθcosθ end fraction end cell row cell         equals space fraction numerator sin to the power of straight 2 straight theta plus cos squared plus 2 sinθcosθ minus 1 over denominator sinθcosθ end fraction end cell row cell from straight space box enclose 2       end cell row cell          rightwards double arrow straight space fraction numerator 1 plus 2 sinθcosθ minus 1 over denominator sinθcosθ end fraction    end cell row cell          rightwards double arrow   2 end cell row cell space So comma space the space correct space option space is space left parenthesis straight c right parenthesis. space space space end cell end table
end style

Question 8

begin mathsize 11px style table attributes columnalign left end attributes row cell left parenthesis secA plus tanA right parenthesis left parenthesis 1 minus sinA right parenthesis equals end cell row cell left parenthesis straight a right parenthesis text  secA      end text left parenthesis straight b right parenthesis text  sinA      end text left parenthesis straight c right parenthesis text  cosecA      end text left parenthesis straight d right parenthesis cosA end cell end table end style

Solution 8

begin mathsize 11px style table attributes columnalign left end attributes row cell text     end text left parenthesis secA plus tanA right parenthesis left parenthesis 1 minus sinA right parenthesis end cell row cell equals text   end text open parentheses 1 over cosA plus sinA over cosA close parentheses left parenthesis 1 minus sinA right parenthesis end cell row cell equals text   end text fraction numerator left parenthesis 1 plus sinA right parenthesis left parenthesis 1 minus sinA right parenthesis over denominator cosA end fraction end cell row cell equals text   end text fraction numerator 1 minus sin squared straight A over denominator cos squared end fraction end cell row cell equals text   end text fraction numerator cos squared straight A over denominator cosA end fraction end cell row cell equals text   cosA end text end cell end table
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 9

begin mathsize 11px style table attributes columnalign left end attributes row cell fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction text  is equal to end text end cell row cell left parenthesis straight a right parenthesis text  sec end text to the power of text 2 end text end exponent straight A text      end text left parenthesis straight b right parenthesis text  -1    end text left parenthesis straight c right parenthesis text  cot end text to the power of text 2 end text end exponent straight A text    end text left parenthesis straight d right parenthesis text  tan end text to the power of text 2 end text end exponent straight A text    end text end cell end table end style

Solution 9

begin mathsize 11px style We space know
table attributes columnalign left end attributes row cell 1 plus tan squared straight A equals sec squared straight A equals fraction numerator 1 over denominator cos squared straight A end fraction text        --- end text box enclose 1 end cell row cell 1 plus cot squared straight A equals cosec squared straight A equals fraction numerator 1 over denominator sin squared straight A end fraction text        --- end text box enclose 1 end cell row cell To text  find:  end text fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction end cell row cell from text    end text box enclose 1 text  &  end text box enclose 2 end cell row cell text      end text rightwards double arrow text   end text fraction numerator fraction numerator 1 over denominator cos squared straight A end fraction over denominator fraction numerator 1 over denominator sin squared straight A end fraction end fraction text   end text end cell row cell text      end text rightwards double arrow text   end text fraction numerator sin squared straight A over denominator cos squared straight A end fraction end cell row cell text      end text rightwards double arrow text  tan end text to the power of text 2 end text end exponent straight A end cell row cell text  So, the correct option is (d). end text end cell end table end style

Question 10

If sin θ - cos θ = 0, then the value of sin4 θ + cos4 θ is

 

  1. 1
Solution 10

Question 11

The value of sin (45° + θ) - cos (45° - θ) is equal to

 

  1. 2 cos θ
  2. 0
  3. 2 sin θ
  4. 1
Solution 11

Question 12

If ΔABC is right angled at C, then the value of cos (A + B) is

 

  1. 0
  2. 1
Solution 12

Chapter 11 - Trigonometric Identities Excercise 11.59

Question 1

If cos 8θ = sin θ and 8θ < 90°, then the value of tan 6θ is

  1. 1
  2. 0
Solution 1

Question 2

If cos (α + β) = 0, then sin (α  -  β) can be reduced to

 

  1. cos β
  2. cos
  3. sin α
  4. sin 2α
Solution 2