If $0 < x < \frac{1}{2}$ and $\alpha=\sin ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^2}}{2}\right)$, then $\tan \alpha+\cot \alpha$ is equal to

If $$x=\sin \left(2 \tan ^{-1} 2\right), y=\cos \left(2 \tan ^{-1} 3\right)$$ and $$z=\sec \left(3 \tan ^{-1} 4\right)$$, then

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

If $$y=\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)$$, where $$x^2 \leq 1$$. Then, find $$\frac{d y}{d x}$$ is equal to

If $$\int \frac{d x}{x\left(\sqrt{\left.x^4-1\right)}\right.}=\frac{1}{k} \sec ^{-1}\left(x^k\right)$$, then the value of $$k$$ is equal to