Simplify

$cos to the power of negative 1 end exponent open square brackets square root of fraction numerator 1 plus cos space 3 x over denominator 1 minus cos space 3 x end fraction end root close square brackets$

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Solution

Let space space space space straight z equals cot to the power of negative 1 end exponent open square brackets square root of fraction numerator 1 plus cos space 3 straight x over denominator 1 minus cos space 3 straight x end fraction end root close square brackets therefore space space space space straight z equals cot to the power of negative 1 end exponent open square brackets square root of fraction numerator 2 space cos squared begin display style 3 over 2 end style straight x over denominator 2 space sin squared space begin display style 3 over 2 end style straight x end fraction end root close square brackets equals cot to the power of negative 1 end exponent open parentheses cot 3 over 2 straight x close parentheses equals 3 over 2 straight x

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