tan to the power of minus 1 end exponent left parenthesis fraction numerator cos x over denominator 1 plus sin x end fraction right parenthesis tan to the power of minus 1 space space end exponent left parenthesis fraction numerator sin x over denominator 1 plus cos x end fraction right parenthesis cos to the power of minus 1 end exponent left parenthesis fraction numerator 1 minus x to the power of 2 n end exponent over denominator 1 plus x to the power of 2 n end exponent end fraction right parenthesis

Asked by Sagar krishna | 21st Jun, 2014, 05:56: PM

Expert Answer:

1.
tan to the power of minus 1 end exponent open parentheses fraction numerator cos x over denominator 1 plus sin x end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator sin open parentheses begin display style straight pi over 2 end style minus x close parentheses over denominator 1 plus cos open parentheses straight pi over 2 minus x close parentheses end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator 2 sin open parentheses begin display style straight pi over 4 end style minus begin display style x over 2 end style close parentheses cos open parentheses straight pi over 4 minus begin display style x over 2 end style close parentheses over denominator 2 cos squared open parentheses straight pi over 4 minus x over 2 close parentheses end fraction close parentheses equals equals tan to the power of minus 1 end exponent open parentheses tan open parentheses straight pi over 4 minus x over 2 close parentheses close parentheses equals open parentheses straight pi over 4 minus x over 2 close parentheses
2.
tan to the power of minus 1 end exponent open parentheses fraction numerator sin x over denominator 1 plus cos x end fraction close parentheses equals tan to the power of minus 1 end exponent open parentheses fraction numerator 2 sin open parentheses x over 2 close parentheses cos open parentheses x over 2 close parentheses over denominator 2 cos squared open parentheses x over 2 close parentheses end fraction close parentheses equals equals tan to the power of minus 1 end exponent open parentheses tan open parentheses x over 2 close parentheses close parentheses equals open parentheses x over 2 close parentheses
3.
W e space n e e d space t o space f i n d space cos to the power of minus 1 end exponent open parentheses fraction numerator 1 minus x to the power of 2 n end exponent over denominator 1 plus x to the power of 2 n end exponent end fraction close parentheses L e t space y equals cos to the power of minus 1 end exponent open parentheses fraction numerator 1 minus x to the power of 2 n end exponent over denominator 1 plus x to the power of 2 n end exponent end fraction close parentheses A s s u m e space t h a t space x to the power of n equals tan theta rightwards double arrow y equals cos to the power of minus 1 end exponent open parentheses fraction numerator 1 minus tan squared theta over denominator 1 plus tan squared theta end fraction close parentheses rightwards double arrow y equals cos to the power of minus 1 end exponent open parentheses cos 2 theta close parentheses rightwards double arrow y equals 2 theta rightwards double arrow y equals 2 tan to the power of minus 1 end exponent open parentheses x to the power of n close parentheses

Answered by Vimala Ramamurthy | 23rd Jun, 2014, 02:50: PM