Find the value of:

$\sin \left({\cos }^{-1}\left(\frac{3}{5}\right)\right)$

(Write answer in fraction form)

Find the value of:

$\tan \left({\mathrm{cosec}}^{-1}\left(\frac{65}{63}\right)\right)$

(Write answer in fraction form)

Find the value of:

$\sin \left(2{\sin }^{-1}\left(\frac{4}{5}\right)\right)$

(Write answer in fraction form)

If the value of: ${\sin }^{-1}\left(\sin \left(\frac{33\mathrm{\pi }}{7}\right)\right)$ is of the form of $\frac{m\mathrm{\pi }}{7}$, then the value of $m=$

If the value of: ${\cos }^{-1}\left(\cos \left(\frac{17\mathrm{\pi }}{6}\right)\right)$ is of the form of $\frac{m\mathrm{\pi }}{6}$, then the value of $m=$

Simplify the following:

${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$

Simplify the following:

${\tan }^{-1}\left(\sec x+\tan x\right)$

Simplify the following:

${\tan }^{-1}\sqrt{\frac{1-\cos x}{1+\cos x}}$

Simplify the following:

${\sin }^{-1}\left(2{\cos }^2\theta -1\right)+{\cos }^{-1}\left(1-2{\sin }^2\theta \right)$