Question

Write the following in the simplest form :

$cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of straight x squared minus 1 end root end fraction close parentheses comma space open vertical bar straight x close vertical bar greater than 1$

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Solution

Put x = sec θ

therefore space space space space cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of straight x squared minus 1 end root end fraction close parentheses equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of sec squared space straight theta minus 1 end root end fraction close parentheses equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator tan space straight theta end fraction close parentheses 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 space space space space equals space cot to the power of negative 1 end exponent left parenthesis cot space straight theta right parenthesis equals straight theta equals sec to the power of negative 1 end exponent straight x therefore space space space cot to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of straight x squared minus 1 end root end fraction close parentheses equals sec to the power of negative 1 end exponent straight x

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