p r o v e space t h a t space f o r space space tan to the power of minus 1 end exponent open parentheses 1 fourth close parentheses plus space tan to the power of minus 1 end exponent open parentheses 2 over 9 close parentheses space equals space 1 half cos to the power of minus 1 end exponent open parentheses 3 over 5 close parentheses

Asked by Lochana Jayachandran | 5th Oct, 2014, 07:57: PM

Expert Answer:

W e space k n o w space t h a t tan to the power of minus 1 end exponent x plus tan to the power of minus 1 end exponent y equals tan to the power of minus 1 end exponent open parentheses fraction numerator x plus y over denominator 1 minus x y end fraction close parentheses T h u s comma space L H S equals tan to the power of minus 1 end exponent open parentheses 1 fourth close parentheses plus tan to the power of minus 1 end exponent open parentheses 2 over 9 close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator 1 fourth plus 2 over 9 over denominator 1 minus 1 fourth cross times 2 over 9 end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator begin display style 17 over 36 end style over denominator 1 minus 1 over 18 end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator begin display style 17 over 36 end style over denominator 17 over 18 end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses 17 over 36 cross times 18 over 17 close parentheses L H S equals tan to the power of minus 1 end exponent open parentheses 1 half close parentheses... left parenthesis 1 right parenthesis A l s o space w e space k n o w space t h a t space tan to the power of minus 1 end exponent x equals cos to the power of minus 1 end exponent open parentheses fraction numerator 1 over denominator square root of 1 plus x squared end root end fraction close parentheses T h u s comma space e q u a t i o n space left parenthesis 1 right parenthesis space b e c o m e s comma L H S equals cos to the power of minus 1 end exponent open parentheses fraction numerator 1 over denominator square root of 1 plus open parentheses begin display style 1 half end style close parentheses squared end root end fraction close parentheses L H S equals cos to the power of minus 1 end exponent open parentheses fraction numerator 2 over denominator square root of 5 end fraction close parentheses L e t space theta equals cos to the power of minus 1 end exponent open parentheses fraction numerator 2 over denominator square root of 5 end fraction close parentheses rightwards double arrow cos theta equals fraction numerator 2 over denominator square root of 5 end fraction W e space k n o w space t h a t space cos 2 theta equals 2 cos squared theta minus 1 T h u s space cos 2 theta equals 2 open parentheses fraction numerator 2 over denominator square root of 5 end fraction close parentheses squared minus 1 rightwards double arrow cos 2 theta equals 2 open parentheses 4 over 5 close parentheses minus 1 rightwards double arrow cos 2 theta equals open parentheses 8 over 5 close parentheses minus 1 rightwards double arrow cos 2 theta equals 3 over 5 rightwards double arrow 2 theta equals cos to the power of minus 1 end exponent open parentheses 3 over 5 close parentheses rightwards double arrow theta equals 1 half cos to the power of minus 1 end exponent open parentheses 3 over 5 close parentheses T h u s comma space L H S equals R H S H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 6th Oct, 2014, 10:57: AM