If $2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x)$, then the value of $x$ is

If $\tan ^{-1}\left(\frac{x}{2}\right)+\tan ^{-1}\left(\frac{y}{2}\right)+\tan ^{-1}\left(\frac{z}{2}\right)=\frac{\pi}{2} \quad$ then $x y+y z+z x=$

$$ \begin{aligned} &\text { The value of }\\ &\begin{aligned} \sin ^{-1}\left(-\frac{1}{\sqrt{2}}\right)+\cos ^{-1} & \left(-\frac{1}{2}\right) -\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\tan ^{-1}(-\sqrt{3}) \text { is } \end{aligned} \end{aligned} $$

If $a^2+b^2+c^2=r^2$, then the value of $\tan ^{-1}\left(\frac{\mathrm{ab}}{\mathrm{cr}}\right)+\tan ^{-1}\left(\frac{\mathrm{bc}}{\mathrm{ar}}\right)+\tan ^{-1}\left(\frac{\mathrm{ca}}{\mathrm{br}}\right)=$