Inverse Trigonometric Functions

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Question

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

Solution

Put x = a sin θ

therefore space space tan to the power of negative 1 end exponent space open parentheses fraction numerator straight x over denominator square root of straight a squared minus straight x squared end root end fraction close parentheses equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight a space sin space straight theta over denominator square root of straight a squared minus straight a squared space sin squared space straight theta end root end fraction close parentheses equals tan to the power of negative 1 end exponent space square root of 1 minus straight z squared end root space space space space equals space tan to the power of negative 1 end exponent left parenthesis tan space straight theta right parenthesis equals straight theta space space space space equals space sin to the power of negative 1 end exponent straight x over straight a space space space space space space space space space space space space space space space space space space space space open square brackets because space straight x equals straight a space sin space straight theta comma space space therefore space sin space straight theta space equals straight x over straight a space space rightwards double arrow space straight theta equals sin to the power of negative 1 end exponent straight x over straight a close square brackets therefore space space space tan to the power of negative 1 end exponent space open parentheses fraction numerator straight x over denominator square root of straight a squared minus straight x squared end root end fraction close parentheses equals sin to the power of negative 1 end exponent straight x over straight a

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