Mathematics : Inverse Trigonometric Functions

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

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

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

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

Prove the following: 

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

Prove the following:

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

Write the value of  sin $\left[ \frac{\mathrm{\pi }}{3} - {\sin }^{-1} \left( - \frac{1}{2}\right) \right]$

Prove the following:

${\cot }^{-1 }\left[ \frac{\sqrt{ 1 + \sin \mathrm{x}} + \sqrt{ 1 - \sin \mathrm{x}}}{\sqrt{ 1 + \sin \mathrm{x}} - \sqrt{ 1 - \sin \mathrm{x}}} \right] = \frac{\mathrm{x}}{2}, \mathrm{x} \in \left( 0, \frac{\mathrm{\pi }}{4} \right)$

Find the value of  ${\tan }^{-1} \left( \frac{\mathrm{x}}{\mathrm{y}} \right) - {\tan }^{-1} \left(\frac{\mathrm{x} - \mathrm{y}}{\mathrm{x} + \mathrm{y}}\right)$

Write the principal value of  ${\cos }^{-1} \left( \frac{1}{2} \right) - 2 {\sin }^{-1} \left( - \frac{1}{2} \right).$