Write the following in the simplest form :

$tan to the power of negative 1 end exponent open square brackets square root of fraction numerator 1 minus cos space straight x over denominator 1 plus sin space straight x end fraction end root close square brackets$

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Solution

tan to the power of negative 1 end exponent open square brackets square root of fraction numerator 1 minus cos space straight x over denominator 1 plus sin space straight x end fraction end root close square brackets equals tan to the power of negative 1 end exponent open square brackets square root of fraction numerator 2 sin squared begin display style x over 2 end style over denominator 2 space cos squared space begin display style x over 2 end style end fraction end root close square brackets equals tan to the power of negative 1 end exponent open square brackets square root of tan squared x over 2 end root close square brackets 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 tan to the power of negative 1 end exponent open parentheses tan x over 2 close parentheses equals x over 2

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