If $\sec^{-1} \left( \frac{x^2 + y^2}{x^2 - y^2} \right) = 2a$, such that $y \frac{dy}{dx} = x \cdot f(a)$, then the value of $f \left( \frac{2\pi}{3} \right)$ is

Let the maximum value of $(\sin^{-1}x)^{2} + (\cos^{-1}x)^{2}$ for $x \in [-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}]$ be $\frac{m}{n}\pi^{2}$, where $\gcd(m, n) = 1$. Then $m+n$ is equal to ........... .

The differential coefficient of $\cos^{-1} \left( \sqrt{\frac{1+x}{2}} \right)$ with respect to $x$ is

Evaluate: $\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]+\cot ^{-1}(-\sqrt{3})$

The value of $x$ which satisfies $\sin \left(\cot ^{-1} x\right)=\cos \left(\tan ^{-1}(1+x)\right)$ is

If $\alpha = \cos^{-1}\left(\frac{3}{5}\right)$ and $\beta = \tan^{-1}\left(\frac{1}{3}\right)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\alpha - \beta$ is equal to