If $X=\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]+\cos ^{-1}\left[\cos \left(\frac{7 \pi}{6}\right)\right]$ and $Y=\sin ^{-1}\left[\sin \left(\frac{11 \pi}{6}\right)\right]+\tan ^{-1}\left[\tan \left(\frac{4 \pi}{3}\right)\right]$ then the value of $\mathbf{2} \boldsymbol{X}-\boldsymbol{Y}$ is:

Which of the following is the simplest form of the expression $\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\sqrt{\mathbf{1 + x ^ { \mathbf { 2 } }}}-\mathbf{1}}{\boldsymbol{x}}\right)$ where $x \neq 0$

The value of the expression $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)+\sin ^{-1}\left(\sin \frac{22 \pi}{3}\right)+\tan ^{-1}\left(\tan \frac{4 \pi}{5}\right)$ is:

$\sin ^{-1}(x-1)+\cos ^{-1}(x-3)+\tan ^{-1}\left(\frac{x}{2-x^2}\right)=\cos ^{-1} k+\pi$, then the value of ' $k$ ' is