Solve $\sin(\tan^{-1} x)$,where $|x| < 1$,is equal to:

If $\frac{\pi}{2} \le x \le \frac{3\pi}{2},$ then $\sin^{-1}(\sin x)$ is equal to

The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is

$\sin ^{-1}\left(\frac{1}{2}\right)+\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)+\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)=$

If $\frac{3x+1}{(x-1)(x+3)}=\frac{A}{x-1}+\frac{B}{x+3}$,then $\sin^{-1} \frac{A}{B}$ is equal to

The value of $\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)$ in the interval $\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$ equals