Question

Prove $2 space tan to the power of negative 1 end exponent space 1 over straight x equals sin to the power of negative 1 end exponent space fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction$

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Solution

Put space 1 over straight x equals tan space straight theta straight L. straight H. straight S. space equals space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 space straight x over denominator straight x squared plus 1 end fraction close parentheses equals sin to the power of negative 1 end exponent open parentheses fraction numerator begin display style 2 over straight x end style over denominator 1 plus begin display style 1 over straight x squared end style end fraction close parentheses equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 space tan space straight theta over denominator 1 plus space tan squared space straight theta end fraction close parentheses equals sin to the power of negative 1 end exponent left parenthesis sin space 2 space straight theta right parenthesis equals 2 space straight theta therefore space space space straight L. straight H. straight S. space equals space straight R. straight H. straight S.

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