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Asked by Kusum and Sanjeet 10th May 2016, 7:00 AM
Answered by Expert
Answer:
1.
G i v e n space t h a t comma space s i n to the power of negative 1 end exponent x plus s i n to the power of negative 1 end exponent y plus s i n to the power of negative 1 end exponent z equals straight pi
L e t space s i n to the power of negative 1 end exponent straight x equals straight A comma space s i n to the power of negative 1 end exponent straight y equals straight B comma space s i n to the power of negative 1 end exponent straight z equals straight C comma space rightwards double arrow s i n A equals straight x comma space s i n B equals straight y comma space s i n C equals straight z
rightwards double arrow straight A plus straight B plus straight C equals straight pi
rightwards double arrow straight A plus straight B equals straight pi minus straight C
rightwards double arrow s i n open parentheses straight A plus straight B close parentheses equals s i n open parentheses straight pi minus straight C close parentheses

C o n s i d e r space
s i n 2 straight A plus s i n 2 straight B plus s i n 2 straight C
equals s i n open parentheses straight pi minus 2 straight A close parentheses plus s i n open parentheses straight pi minus 2 straight B close parentheses plus s i n open parentheses straight pi minus 2 straight C close parentheses
equals 2 s i n open parentheses straight A plus straight B close parentheses c o s open parentheses straight A minus straight B close parentheses plus 2 s i n C c o s C
equals 2 s i n C c o s open parentheses straight A minus straight B close parentheses plus 2 s i n C c o s C
equals 2 s i n C open square brackets c o s open parentheses straight A minus straight B close parentheses plus c o s C close square brackets
equals 2 s i n C open square brackets c o s open parentheses straight A minus straight B close parentheses plus c o s open parentheses straight A plus straight B close parentheses close square brackets
equals 2 s i n C times 2 s i n A times s i n B
equals 4 s i n A s i n B s i n C
rightwards double arrow 2 sin A cos A plus 2 sin B cos B plus 2 sin C cos C equals 4 s i n A s i n B s i n C
rightwards double arrow s i n A c o s A plus s i n B c o s B plus s i n C c o s C equals 2 s i n A s i n B s i n C
rightwards double arrow s i n A cross times square root of 1 minus sin squared A end root plus s i n B cross times square root of 1 minus s i n squared B end root plus s i n C cross times square root of 1 minus s i n squared C end root equals 2 s i n A s i n B s i n C
rightwards double arrow x cross times square root of 1 minus x squared end root plus y cross times square root of 1 minus y squared end root plus z cross times square root of 1 minus z squared end root equals 2 x y z

2.
G i v e n space t h a t comma space s i n to the power of negative 1 end exponent x plus s i n to the power of negative 1 end exponent y plus s i n to the power of negative 1 end exponent z equals straight pi
rightwards double arrow space s i n to the power of negative 1 end exponent straight x plus s i n to the power of negative 1 end exponent straight y equals straight pi minus s i n to the power of negative 1 end exponent straight z
rightwards double arrow c o s space open parentheses s i n to the power of negative 1 end exponent straight x plus s i n to the power of negative 1 end exponent straight y close parentheses equals c o s open parentheses straight pi minus s i n to the power of negative 1 end exponent straight z close parentheses
rightwards double arrow c o s space open parentheses s i n to the power of negative 1 end exponent straight x close parentheses c o s open parentheses s i n to the power of negative 1 end exponent straight y close parentheses minus s i n space open parentheses s i n to the power of negative 1 end exponent straight x close parentheses s i n open parentheses s i n to the power of negative 1 end exponent straight y close parentheses equals negative c o s open parentheses s i n to the power of negative 1 end exponent straight z close parentheses

W e space k n o w space t h a t comma space c o s open parentheses s i n to the power of negative 1 end exponent straight x close parentheses equals c o s open parentheses c o s to the power of negative 1 end exponent square root of 1 minus straight x squared end root close parentheses
T h u s comma
c o s open parentheses c o s to the power of negative 1 end exponent square root of 1 minus straight x squared end root close parentheses c o s open parentheses c o s to the power of negative 1 end exponent square root of 1 minus straight y squared end root close parentheses minus s i n space open parentheses s i n to the power of negative 1 end exponent straight x close parentheses s i n open parentheses s i n to the power of negative 1 end exponent straight y close parentheses equals negative c o s open parentheses s i n to the power of negative 1 end exponent straight z close parentheses space
rightwards double arrow square root of 1 minus straight x squared end root square root of 1 minus straight y squared end root minus x y equals negative square root of 1 minus straight z squared end root
rightwards double arrow square root of 1 minus straight x squared end root square root of 1 minus straight y squared end root equals x y minus square root of 1 minus straight z squared end root

S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma
open parentheses 1 minus straight x squared close parentheses open parentheses 1 minus straight y squared close parentheses equals straight x squared straight y squared plus 1 minus straight z squared minus 2 x y square root of 1 minus straight z squared end root
straight x squared plus straight y squared minus straight z squared equals 2 x y square root of 1 minus straight z squared end root

S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma
open parentheses straight x squared plus straight y squared minus straight z squared close parentheses squared equals 4 straight x squared straight y squared open parentheses 1 minus straight z squared close parentheses
rightwards double arrow straight x to the power of 4 plus straight y to the power of 4 plus straight z to the power of 4 plus 2 straight x squared straight y squared minus 2 straight y squared straight z squared minus 2 straight x squared straight z squared equals 4 straight x squared straight y squared minus 4 straight x squared straight y squared straight z squared
rightwards double arrow straight x to the power of 4 plus straight y to the power of 4 plus straight z to the power of 4 plus 4 straight x squared straight y squared straight z squared equals 4 straight x squared straight y squared minus 2 straight x squared straight y squared plus 2 straight y squared straight z squared plus 2 straight x squared straight z squared
rightwards double arrow straight x to the power of 4 plus straight y to the power of 4 plus straight z to the power of 4 plus 4 straight x squared straight y squared straight z squared equals 2 straight x squared straight y squared plus 2 straight y squared straight z squared plus 2 straight x squared straight z squared
rightwards double arrow straight x to the power of 4 plus straight y to the power of 4 plus straight z to the power of 4 plus 4 straight x squared straight y squared straight z squared equals 2 open parentheses straight x squared straight y squared plus straight y squared straight z squared plus straight x squared straight z squared close parentheses
Answered by Expert 10th May 2016, 9:37 PM
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