Find the value of $\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$.

$\tan \left(2 \tan^{-1}\left(\frac{1}{3}\right)+\tan^{-1}\left(\frac{1}{7}\right)\right) = $

$\tan ^{-1} \sqrt{3} - \cot ^{-1}(-\sqrt{3}) = $ . . . . . . .

Let $f(\theta) = \sin ( \tan ^{-1} ( \frac{\sin \theta}{\sqrt{\cos 2 \theta}} ) )$,where $-\frac{\pi}{4} < \theta < \frac{\pi}{4}$,then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is

The value of $x$ which satisfies $\sin \left(\cot ^{-1} x\right)=\cos \left(\tan ^{-1}(1+x)\right)$ is

If $\cos^{-1} p + \cos^{-1} q + \cos^{-1} r = \pi$,then $p^2 + q^2 + r^2 + 2pqr = $