Inverse Trigonometric Functions

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Question

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

Solution

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