If $\sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3}$, then the value of $\cos^{-1} x + \cos^{-1} y$ is :

Let $\tan^{-1}y = \tan^{-1}x + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Then $y$ is :

The value of $\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]$ is :

$\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}$, than $x$ is equal to

(a) $0$

(b) $1$

(c) $\frac{1}{2}$

(d) $2$

Choose the most appropriate answer from the options given below:

The simplest form of $\tan^{-1} \frac{\sqrt{1+x^2} - 1}{x}$, $x \neq 0$ is:

$\cos[\tan^{-1}\{\sin(\cot^{-1}x)\}]$ is equal to

The maximum value of $\sin x \cdot \cos x$ is: