If $\cos \left(\cos ^{-1} \frac{\sqrt{3}}{2}+\sin ^{-1} x\right)=1$,then find the value of $x$.

Considering the principal values of the inverse trigonometric functions,$\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^2}\right)$,for $-\frac{1}{2} < x < \frac{1}{\sqrt{2}}$,is equal to:

If $ a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b $,then find the values of $ a $ and $ b $.

Find the derivative: $\frac{d}{dx} \cos^{-1} \left( \frac{x - x^{-1}}{x + x^{-1}} \right)$

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

$\begin{aligned} & \text{If } \cot \left(\cos ^{-1} x\right)=\sec \left\{\tan ^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right\} \\ & b>a, \text{ then } x= \end{aligned}$