$$\frac{d}{d x}\left\{\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\right\}$$ is equal to

If $$y=\tan ^{-1}\left\{\frac{a x-b}{b x+a}\right\}$$, then $$y^{\prime}$$ is equal to

For how many distinct values of $$x$$, the following $$\sin \left[2 \cos ^{-1} \cot \left(2 \tan ^{-1} x\right)\right]=0$$ holds?

If $$\tan ^{-1}\left[\frac{1}{1+1 \cdot 2}\right]+\tan ^{-1}\left[\frac{1}{1+2 \cdot 3}\right]+\ldots+\tan ^{-1} \left[\frac{1}{1+n(1+1)}\right]=\tan ^{-1}[x]$$, then $$x$$ is equal to