Prove $tan to the power of negative 1 end exponent square root of x equals 1 half cos to the power of negative 1 end exponent open parentheses fraction numerator 1 minus x over denominator 1 plus x end fraction close parentheses$

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Solution

Put space straight x equals tan squared space straight theta space or space square root of straight x equals tan space straight theta space or space straight theta equals tan to the power of negative 1 end exponent square root of straight x straight R. straight H. straight S. space equals 1 half cos to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses equals 1 half cos to the power of negative 1 end exponent open parentheses fraction numerator 1 minus tan squared space straight theta over denominator 1 plus tan squared space straight theta end fraction close parentheses space space space space space space space space space space space equals 1 half space cos to the power of negative 1 end exponent left parenthesis cos space 2 space straight theta right parenthesis equals 1 half left parenthesis 2 space straight theta right parenthesis equals straight theta equals tan to the power of negative 1 end exponent square root of straight x equals straight L. straight H. straight S.

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