Simplify from Mathematics Inverse Trigonometric Functions
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    Inverse Trigonometric Functions

    Multiple Choice QuestionsShort 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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    Answer
    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
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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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    Answer
    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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    Answer
    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

    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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