The value of $\cos \left[\sec ^{-1} x+\operatorname{cosec}^{-1} x\right], |x| \geq 1$ is equal to: . . . . . . .

If $\cot ^{-1}(\alpha)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18+\cot ^{-1} 32+\ldots$ up to $100$ terms,then $\alpha$ is

If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ is equal to

If $\sin^{-1} x = \frac{\pi}{5}$ for some $x \in (-1, 1)$,then the value of $\cos^{-1} x$ is

If $x$ takes non-positive permissible value,then $\sin^{-1} x =$

Let $\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right) = \alpha$ and $\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right) = \beta$. Then: