Prove that begin mathsize 20px style sin open curly brackets tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus x squared over denominator 2 x end fraction close square brackets plus cos to the power of negative 1 end exponent open square brackets fraction numerator 1 minus x squared over denominator 1 plus x squared end fraction close square brackets close curly brackets equals 1 end style

Asked by sunil2791 | 21st Apr, 2017, 07:31: PM

Expert Answer:

begin mathsize 16px style LHS equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 2 straight x end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style 1 minus straight x squared end style over denominator begin display style 1 plus straight x squared end style end fraction close parentheses close square brackets
Substitute space straight x equals tanθ
sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 2 straight x end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style 1 minus straight x squared end style over denominator begin display style 1 plus straight x squared end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus tan squared straight theta over denominator 2 tanθ end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style 1 minus tan squared straight theta end style over denominator begin display style 1 plus tan squared straight theta end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus fraction numerator sin squared straight theta over denominator cos squared straight theta end fraction over denominator begin display style fraction numerator 2 sinθ over denominator cos straight theta end fraction end style end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style 1 minus fraction numerator sin squared straight theta over denominator cos squared straight theta end fraction end style over denominator begin display style 1 plus fraction numerator sin squared straight theta over denominator cos squared straight theta end fraction end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator fraction numerator cos squared straight theta minus sin squared straight theta over denominator cos squared straight theta end fraction over denominator begin display style 2 sinθ space cosθ end style end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style fraction numerator cos squared straight theta minus sin squared straight theta over denominator cos squared straight theta end fraction end style over denominator begin display style fraction numerator cos squared straight theta plus sin squared straight theta over denominator cos squared straight theta end fraction end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator cos squared straight theta minus sin squared straight theta over denominator begin display style sin 2 straight theta end style end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style cos squared straight theta minus sin squared straight theta end style over denominator begin display style cos squared straight theta plus sin squared straight theta end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator cos 2 straight theta over denominator begin display style sin 2 straight theta end style end fraction close parentheses plus cos to the power of negative 1 end exponent open parentheses fraction numerator begin display style cos 2 straight theta end style over denominator begin display style 1 end style end fraction close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses cot 2 straight theta close parentheses plus cos to the power of negative 1 end exponent open parentheses cos 2 straight theta close parentheses close square brackets
equals sin open square brackets tan to the power of negative 1 end exponent open parentheses tan open parentheses straight pi over 2 minus 2 straight theta close parentheses close parentheses plus 2 straight theta close square brackets
equals sin open square brackets straight pi over 2 minus 2 straight theta plus 2 straight theta close square brackets
equals sin straight pi over 2
equals 1
equals RHS
Hence space proved.

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Answered by Rebecca Fernandes | 27th Nov, 2017, 12:43: PM

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