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Inverse Trigonometric Functions

Short 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

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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.

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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.

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97. If cos ^–1x + cos^–1y + cos^–1z = π prove that x² + y² + z² + 2 x y z = 1.

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98. If sin^–1'x + sin^–1 y + sin^–1 z = π. show that

$straight x square root of 1 minus straight x squared end root plus straight y square root of 1 minus straight y squared end root plus straight z square root of 1 minus straight z squared end root equals 2 space straight x space straight y space straight z$

We are given that
sin^–1x + sin–¹y+sin^–1z = π
Put x = sin A,y = sin B, z = sin C
∴ sin x = A, sin–1 y = B, sin^–1z = C
∴ we have
A + B + C = $straight pi$
$straight L. straight H. straight S. space equals space square root of 1 minus straight x squared end root plus straight y square root of 1 minus straight y to the power of 2 degree end exponent end root plus straight z square root of 1 minus straight z squared end root space space space space space space space space space space equals sin space straight A square root of 1 minus sin squared space straight A end root plus sin space straight B square root of 1 minus sin squared space straight B end root plus sin space straight C square root of 1 minus sin squared space straight C end root space space space space space space space space space space equals space sin space straight A. space cos space straight A space plus space sin space space straight B. space cos space straight B plus space sin space straight C space. space cos space straight C space space space space space space space space space space equals 1 half left square bracket 2 space sin space straight A. cos space straight A plus 2 space sin space straight B. space cos space straight B plus 2 space sin space straight C. space cos space straight C right square bracket space space space space space space space space space space equals 1 half left square bracket left parenthesis sin space 2 space straight A plus space sin space 2 space straight B right parenthesis plus sin space 2 space straight C right square bracket equals 1 half left square bracket 2 space sin space left parenthesis straight A plus straight B right parenthesis cos left parenthesis straight A minus straight B right parenthesis plus sin space 2 space straight C right square bracket space space space space space space space space space space equals space 1 half left square bracket 2 space sin left parenthesis 180 degree minus straight C right parenthesis space cos space left parenthesis straight A minus straight B right parenthesis plus sin space 2 straight C right square bracket space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space equals space 1 half space left square bracket 2 space sin space straight C. space cos space left parenthesis straight A minus straight B right parenthesis plus cos space straight C right curly bracket space space space space space space space space space space equals space sin space straight C left square bracket cos left parenthesis straight A minus straight B right parenthesis plus cos left curly bracket 180 degree minus left parenthesis straight A plus straight B right parenthesis right curly bracket right square bracket space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space equals space sin space straight C left square bracket cos left parenthesis straight A minus straight B right parenthesis minus cos left parenthesis straight A plus straight B right parenthesis right square bracket minus sinC left square bracket 2 space sin space straight A space sin space straight B right square bracket space space space space space space space space space space equals space 2 space sin space straight A space sin space straight B space sin space straight C equals space 2 space xyz space space space space space space space space space space equals space straight R. straight H. straight S.$