Q.1tan to the power of negative 1 end exponent open parentheses square root of x plus close parentheses square root of y divided by 1 minus square root of x y end root right parenthesis equals tan to the power of negative 1 end exponent square root of x plus tan to the power of negative 1 end exponent open parentheses square root of y close parentheses tan to the power of negative 1 end exponent open parentheses x plus square root of x close parentheses divided by 1 minus x to the power of 3 divided by 2 end exponent equals tan to the power of negative 1 end exponent open parentheses x close parentheses plus tan to the power of negative 1 end exponent open parentheses square root of x close parentheses tan to the power of negative 1 end exponent open parentheses sin x divided by 1 plus cos x close parentheses equals x divided by 2

Asked by debjyotisaha98 | 23rd Jul, 2015, 09:41: PM

Expert Answer:

1. L e t space tan to the power of negative 1 end exponent square root of x equals A space a n d space tan to the power of negative 1 end exponent square root of y equals B rightwards double arrow square root of x equals tan A space a n d square root of space y end root equals tan B N o w space c o n s i d e r comma tan open parentheses A plus B close parentheses equals fraction numerator tan A plus tan B over denominator 1 minus tan A cross times tan B end fraction space space space space space space space space space space space space space space space space equals fraction numerator square root of x plus square root of y over denominator 1 minus square root of x square root of y end fraction rightwards double arrow t a n open parentheses t a n to the power of negative 1 end exponent square root of x plus space t a n to the power of negative 1 end exponent square root of y close parentheses equals fraction numerator square root of x plus square root of y over denominator 1 minus square root of x square root of y end fraction rightwards double arrow t a n to the power of negative 1 end exponent square root of x plus space t a n to the power of negative 1 end exponent square root of y equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of x plus square root of y over denominator 1 minus square root of x square root of y end fraction close square brackets.... left parenthesis 1 right parenthesis 2. N o w space s u b s t i t u t i n g space square root of x equals x space a n d space square root of y equals square root of x space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma t a n to the power of negative 1 end exponent x plus space t a n to the power of negative 1 end exponent square root of x equals t a n to the power of negative 1 end exponent open square brackets fraction numerator x plus square root of x over denominator 1 minus x square root of x end fraction close square brackets rightwards double arrow t a n to the power of negative 1 end exponent x plus space t a n to the power of negative 1 end exponent square root of x equals t a n to the power of negative 1 end exponent open square brackets fraction numerator x plus square root of x over denominator 1 minus x to the power of begin display style 3 over 2 end style end exponent end fraction close square brackets.... left parenthesis 2 right parenthesis 3. tan to the power of negative 1 end exponent fraction numerator sin x over denominator 1 plus cos x end fraction equals t a n to the power of negative 1 end exponent fraction numerator 2 s i n begin display style x over 2 end style cross times cos begin display style x over 2 end style over denominator 1 plus 2 c o s squared begin display style x over 2 end style minus 1 end fraction 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 equals t a n to the power of negative 1 end exponent fraction numerator 2 s i n begin display style x over 2 end style cross times c o s begin display style x over 2 end style over denominator 1 plus 2 c o s squared begin display style x over 2 end style minus 1 end fraction 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 equals t a n to the power of negative 1 end exponent fraction numerator 2 s i n begin display style x over 2 end style cross times c o s begin display style x over 2 end style over denominator 2 c o s squared begin display style x over 2 end style end fraction 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 equals t a n to the power of negative 1 end exponent fraction numerator s i n begin display style x over 2 end style over denominator c o s begin display style x over 2 end style end fraction 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 equals t a n to the power of negative 1 end exponent open parentheses tan x over 2 close parentheses 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 equals x over 2

Answered by Vimala Ramamurthy | 24th Jul, 2015, 09:22: AM