$\tan \left[ 2\tan^{-1}\left( \frac{1}{5} \right) - \frac{\pi}{4} \right] = $

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

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

If $\tan ^{-1}\left(\frac{1}{3}\right) + \tan ^{-1}\left(\frac{1}{7}\right) + \tan ^{-1}\left(\frac{1}{13}\right) + \tan ^{-1}\left(\frac{1}{21}\right) + \tan ^{-1}\left(\frac{1}{31}\right) = \tan ^{-1}\left(\frac{p}{q}\right)$,where $p$ and $q$ are relatively prime numbers,then $p + q$ is equal to-

If $2\tan^{-1}(\cos x) = \tan^{-1}(2\csc x)$,then $x =$

If $y = \sin^{-1} \left( \frac{\sqrt{1+x} + \sqrt{1-x}}{2} \right)$,then $\frac{dy}{dx} = $