If $$\sin ^{-1}\left(\frac{2 a}{1+a^2}\right)+\cos ^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)$$ where $$a, x \in(0,1)$$, then the value of $$x$$ is

The value of $$\cot ^{-1}\left[\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right]$$, where $$x \in\left(0, \frac{\pi}{4}\right)$$ is

Domain $$\cos ^{-1}[x]$$ is, where [ ] denotes a greatest integer function

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