Find the principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$.

If $y = \cos^{-1}(\cos(|x| - f(x)))$,where $f(x) = \begin{cases} 1, & \text{if } x > 0 \\ -1, & \text{if } x < 0 \\ 0, & \text{if } x = 0 \end{cases}$,then $\left. \frac{dy}{dx} \right|_{x = \frac{5\pi}{4}}$ is

Evaluate the expression: $sec^2(\tan^{-1} 3) - \tan^2(sec^{-1} 3) = $

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

$\frac{\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos ^{-1}\left(-\frac{1}{2}\right)}=$

$\tan \left( 90^o - \cot^{-1} \frac{1}{3} \right) = $