Write the following in the simplest form : from Mathematics Inv

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Inverse Trigonometric Functions

Short Answer Type

81.

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.

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82.

Write $tan to the power of negative 1 end exponent open square brackets fraction numerator 3 straight a squared space straight x minus straight x cubed over denominator straight a left parenthesis straight a cubed minus 3 straight x squared right parenthesis end fraction close square brackets comma space straight a greater than 0 comma space space minus fraction numerator straight a over denominator square root of 3 end fraction less or equal than straight x less or equal than fraction numerator straight a over denominator square root of 3 end fraction$ in the simplest form.

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

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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85. Write the following in the simplest form :

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

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86. Write the following in the simplest form :

tan to the power of negative 1 end exponent open parentheses fraction numerator cos space straight x minus sin space straight x over denominator cos space straight x plus sin space straight x end fraction close parentheses

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87. 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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88. Simplify

tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus x squared end root minus 1 over denominator x end fraction close parentheses

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89. Simplify

tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus x squared end root minus 1 over denominator x end fraction close parentheses

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90. 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

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