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Inverse Trigonometric Functions

Short Answer Type

121.

Using principal value, evaluate the following: $\sin^{- 1} (\sin \frac{3 π}{5})$

Long Answer Type

122.

Prove that: $\sin^{- 1} (\frac{4}{5}) + \sin^{- 1} (\frac{5}{13}) + \sin^{- 1} (\frac{16}{65}) = \frac{π}{2}$

123.

Solve for x: $\tan^{- 1} 3 x + \tan^{- 1} 2 x = \frac{π}{4}$

Short Answer Type

124.

What is the principal value of $\cos^{- 1} (- \frac{\sqrt{3}}{2})$ ?

Long Answer Type

125.

Prove the following:

$\tan^{- 1} \sqrt{x} = \frac{1}{2} \cos^{- 1} (\frac{1 - x}{1 + x}), x \in (0, 1)$

126.

Prove the following:

$\cos^{- 1} (\frac{12}{13}) + \sin^{- 1} (\frac{3}{5}) = \sin^{- 1} (\frac{56}{65})$

Short Answer Type

127.

Write the value of sin $[\frac{π}{3} - \sin^{- 1} (- \frac{1}{2})]$

Long Answer Type

128.
Prove the following:
$\cot^{- 1} [\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}] = \frac{x}{2}, x \in (0, \frac{π}{4})$

$\cot^{- 1} [\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}] \cot^{- 1} [\frac{\sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + \sin 2 (\frac{x}{2})} + \sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - \sin 2 (\frac{x}{2})}}{\sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + \sin 2 (\frac{x}{2})} - \sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - \sin 2 (\frac{x}{2})}}]$

$[Since, \sin^{2} A + \cos^{2} A = 1] = \cot^{- 1} [\frac{\sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})} + \sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}}{\sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})} - \sqrt{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}}]$

[ Since, sin 2A = 2 sin A cos A ]

$= \cot^{- 1} [\frac{\sqrt{{(\cos \frac{x}{2} + \sin \frac{x}{2})}^{2}} + \sqrt{{(\cos \frac{x}{2} - \sin \frac{x}{2})}^{2}}}{\sqrt{{(\cos \frac{x}{2} + \sin \frac{x}{2})}^{2}} - \sqrt{{(\cos \frac{x}{2} - \sin \frac{x}{2})}^{2}}}] = \cot^{- 1} (\frac{2 \cos \frac{x}{2}}{2 \sin \frac{x}{2}}) = \cot^{- 1} (\cot \frac{x}{2}) = \frac{x}{2}$