$\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right)$ is

The value of $\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)$ is

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

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ where $0 \leq \cos ^{-1} x \leq \pi$ and $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$, is