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Inverse Trigonometric Functions

Question

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Simplify

$cot to the power of negative 1 end exponent open parentheses square root of 1 plus straight x squared end root minus straight x close parentheses$

Solution

Let space space space straight z equals space cot to the power of negative 1 end exponent space open parentheses square root of 1 plus straight x squared end root minus straight x close parentheses Put space space space straight x equals tan space straight theta therefore space space space space straight z equals cot to the power of negative 1 end exponent space open parentheses square root of 1 plus tan squared space straight theta end root minus tan space straight theta close parentheses equals cot to the power of negative 1 end exponent left parenthesis sec space straight theta minus tan space straight theta right parenthesis space space space space space space space space equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator cos space straight theta end fraction minus fraction numerator sin space straight theta over denominator cos space straight theta end fraction close parentheses equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 minus sin space straight theta over denominator cos space straight theta end fraction close parentheses space space space space space space space space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator cos squared begin display style straight theta over 2 end style plus sin squared begin display style straight theta over 2 end style minus 2 space cos begin display style straight theta over 2 end style space sin space begin display style straight theta over 2 end style over denominator cos squared begin display style straight theta over 2 end style minus sin squared begin display style straight theta over 2 end style end fraction close square brackets space space space space space space space space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator open parentheses cos begin display style straight theta over 2 end style minus sin begin display style straight theta over 2 end style close parentheses squared over denominator open parentheses cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style close parentheses space open parentheses cos straight theta over 2 plus sin straight theta over 2 close parentheses end fraction close square brackets space space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator cos begin display style straight theta over 2 end style minus sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style end fraction close square brackets equals cos to the power of negative 1 end exponent open square brackets fraction numerator 1 minus tan begin display style straight theta over 2 end style over denominator 1 plus tan begin display style straight theta over 2 end style end fraction close square brackets space equals space cot to the power of minus open square brackets cot open parentheses straight pi over 4 plus straight theta over 2 close parentheses close square brackets equals straight pi over 4 plus straight theta over 2 equals straight pi over 4 plus 1 half tan to the power of negative 1 end exponent straight x

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