. from Mathematics Inverse Trigonometric Functions
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    Inverse Trigonometric Functions

    Multiple Choice QuestionsShort Answer Type

    91. Simplify

    cot to the power of negative 1 end exponent open parentheses square root of 1 plus straight x squared end root plus straight x close parentheses
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    92. Prove that tan to the power of negative 1 end exponent x plus tan to the power of negative 1 end exponent fraction numerator 2 space x over denominator 1 minus x squared end fraction equals tan to the power of negative 1 end exponent open parentheses fraction numerator 3 x minus x cubed over denominator 1 minus 3 x squared end fraction close parentheses equals tan to the power of negative 1 end exponent open parentheses fraction numerator 3 x minus x cubed over denominator 1 minus 3 x squared end fraction close parentheses comma open vertical bar x close vertical bar less than fraction numerator 1 over denominator square root of 3 end fraction.
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    93. Show space that space tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets equals straight pi over 4 plus 1 half cos to the power of negative 1 end exponent space straight x squared.

    Put x2 = cos θ

    straight L. straight H. straight S. space equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root plus square root of 1 minus straight x squared end root over denominator square root of 1 plus straight x squared end root minus square root of 1 minus straight x squared end root end fraction close square brackets equals tan to the power of negative 1 end exponent space open square brackets fraction numerator square root of 1 plus cos space straight theta end root plus square root of 1 minus cos space straight theta end root over denominator square root of 1 plus cos space straight theta end root minus square root of 1 minus cos space straight theta end root end fraction close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent space open square brackets fraction numerator square root of 2 space cos squared begin display style straight theta over 2 end style end root plus square root of 2 space sin squared begin display style straight theta over 2 end style end root over denominator square root of 2 space cos squared begin display style straight theta over 2 end style end root minus square root of 2 space sin squared begin display style straight theta over 2 end style end root end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 cos begin display style straight theta over 2 end style plus square root of 2 sin begin display style straight theta over 2 end style over denominator square root of 2 cos begin display style straight theta over 2 end style minus square root of 2 sin begin display style straight theta over 2 end style end fraction close square brackets space space space space space space space space space space equals space tan to the power of negative 1 end exponent open square brackets fraction numerator cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style over denominator cos straight theta over 2 minus sin straight theta over 2 end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 plus tan begin display style straight theta over 2 end style over denominator 1 minus tan begin display style straight theta over 2 end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 plus straight theta over 2 close parentheses close square brackets space space space space space space space space space space equals space straight pi over 4 plus straight theta over 2 equals straight pi over 4 plus 1 half cos to the power of negative 1 end exponent straight x squared 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 space space space space space space space space space space left square bracket because space cos space straight theta equals straight x squared right square bracket space space space space space space space space space space equals straight R. straight H. straight S.
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    94. Prove space that space cot to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus sin space straight x end root plus square root of 1 minus sin space straight x end root over denominator square root of 1 plus sin space straight x end root minus square root of 1 minus sin space straight x end root end fraction close square brackets equals straight x over 2 comma space space straight x space element of space open parentheses 0 comma space straight pi over 4 close parentheses.
    108 Views
    Answer
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    95. If tan–1.x + tan–1y + tan –1z = straight pi, prove that x + y + z = x y z.
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    96. If space cos to the power of negative 1 end exponent straight x over straight a plus cos to the power of negative 1 end exponent straight y over straight b equals straight a comma space space prove space that space straight x squared over straight a squared minus fraction numerator 2 space straight x space straight y over denominator straight a space straight b end fraction cos space straight a plus straight y squared over straight b squared equals sini squared straight a.
    138 Views
    Answer
    97. If cos –1x + cos–1y + cos–1z = π prove that x2 + y2 + z2 + 2 x y z = 1.
    3839 Views
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    98. If sin–1'x + sin–1 y + sin–1 z = π. show that

    straight x square root of 1 minus straight x squared end root plus straight y square root of 1 minus straight y squared end root plus straight z square root of 1 minus straight z squared end root equals 2 space straight x space straight y space straight z
    623 Views
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    Multiple Choice QuestionsMultiple Choice Questions

    99. cos to the power of negative 1 end exponent open parentheses cos fraction numerator 7 space straight pi over denominator 6 end fraction close parentheses space is equal to
    • fraction numerator 7 straight pi over denominator 6 end fraction
    • fraction numerator 5 straight pi over denominator 6 end fraction
    • straight pi over 3
    • straight pi over 3
    142 Views
    Answer
    100. sin open square brackets straight pi over 3 minus sin to the power of negative 1 end exponent open parentheses negative 1 half close parentheses close square brackets is equal to
    • 1 half
    • 1 third
    • 1 fourth
    • 1 fourth
    162 Views
    Answer
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