Evaluate: $\cos(\cos^{-1}(-\frac{1}{2}) + \frac{\pi}{3}) = $

If $\sin ^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4}-\ldots \infty\right) + \cos ^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\ldots \infty\right)=\frac{\pi}{2}$ and $0 < x < \sqrt{2}$,then $x$ is equal to

For $\theta \in \left(0, \frac{\pi}{2}\right)$, $\operatorname{sech}^{-1}(\cos \theta)$ is equal to

$\sin ^{-1}\left(\frac{3}{5}\right)-\sin ^{-1}\left(\frac{8}{17}\right)=$ . . . . . .

$S = \tan^{-1}\left( \frac{1}{n^2 + n + 1} \right) + \tan^{-1}\left( \frac{1}{n^2 + 3n + 3} \right) + \dots + \tan^{-1}\left( \frac{1}{1 + (n + 19)(n + 20)} \right)$,then $\tan S$ is equal to

If $\sec ^{-1} \frac{x}{a}-\sec ^{-1} \frac{x}{b}=\sec ^{-1} b-\sec ^{-1} a$,then $x$ is equal to