$$\cos \left[\cot ^{-1}(-\sqrt{3})+\frac{\pi}{6}\right]$$ is equal to

$$\tan ^{-1}\left[\frac{1}{\sqrt{3}} \sin \frac{5 \pi}{2}\right] \sin ^{-1}\left[\cos \left(\sin ^{-} \frac{\sqrt{3}}{2}\right)\right]$$ is equal to

If $$f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$$, then $$f^{\prime}(\sqrt{3})$$ is

The domain of the function defined by $$f(x)=\cos ^{-1} \sqrt{x-1}$$ is