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Inverse Trigonometric Functions

Short Answer Type

111.

Write the value of $open parentheses 2 tan to the power of negative 1 end exponent 1 fifth close parentheses$

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112.

Find the value of the following:
$tan 1 half open square brackets sin to the power of negative 1 end exponent fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction plus cos to the power of negative 1 end exponent fraction numerator 1 minus straight y squared over denominator 1 plus straight y squared end fraction close square brackets comma space open vertical bar straight x close vertical bar space less than 1 comma space space straight y greater than 0 space and space xy less than 1$

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113.

Prove that $tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 fifth close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space equals straight pi over 4$

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114.

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of the z-axis.

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115.

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116.

Prove that

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117.

Find the value of $\tan^{- 1} \sqrt{3} - c o t^{- 1} (- \sqrt{3})$

118.

Prove that: $3 \sin^{- 1} x = \sin^{- 1} (3 x - 4 x^{3}), X \in [- \frac{1}{2}, \frac{1}{2}]$

119.

Evaluate: $\sin [\frac{π}{3} - \sin^{- 1} (\frac{1}{2})]$

Long Answer Type

120.
Prove the following:
$\tan^{- 1} \frac{1}{3} + \tan^{- 1} \frac{1}{5} + \tan^{- 1} \frac{1}{7} + \tan^{- 1} \frac{1}{8}$

$L . H . S . = \tan^{- 1} (\frac{1}{3}) + \tan^{- 1} (\frac{1}{5}) + \tan^{- 1} (\frac{1}{7}) + \tan^{- 1} (\frac{1}{8}) = \tan^{- 1} (\frac{\frac{1}{3} + \frac{1}{5}}{1 - (\frac{1}{3}) (\frac{1}{5})}) + \tan^{- 1} (\frac{\frac{1}{7} + \frac{1}{8}}{1 - (\frac{1}{7}) (\frac{1}{8})}) = \tan^{- 1} [\frac{\frac{5 + 3}{15}}{\frac{15 - 1}{15}}] + \tan^{- 1} [\frac{\frac{8 + 7}{56}}{\frac{56 - 1}{56}}] = \tan^{- 1} [\frac{\frac{8}{15}}{\frac{14}{15}}] + \tan^{- 1} [\frac{\frac{15}{56}}{\frac{55}{56}}] = \tan^{- 1} (\frac{8}{14}) + \tan^{- 1} (\frac{15}{55}) = \tan^{- 1} (\frac{4}{7}) + \tan^{- 1} (\frac{3}{11}) = \tan^{- 1} (\frac{\frac{4}{7} + \frac{3}{11}}{1 - (\frac{4}{7}) (\frac{3}{11})}) = \tan^{- 1} (\frac{\frac{44 + 21}{77}}{\frac{77 - 12}{77}}) = \tan^{- 1} (\frac{65}{65}) = \tan^{- 1} 1 (\tan \frac{π}{4}) = \frac{π}{4}$