111.

Write the value of 

Find the value of the following:

Prove that 

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.

Prove that 

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

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

Evaluate:   $\sin \left[\frac{\mathrm{\pi }}{3} - {\sin }^{-1} \left(\frac{1}{2}\right)\right]$

Prove the following:

${\tan }^{-1}\frac{1}{3} + {\tan }^{-1}\frac{1}{5} + {\tan }^{-1}\frac{1}{7} + {\tan }^{-1}\frac{1}{8}$