Inverse Trigono...

Number of solutions of the equation $\sin \left ( \displaystyle \frac{1}{3}\cos^{-1}x \right )=1$ are

If value of $x$ which satisfy equation $\displaystyle \sin^{-1}\frac{2x}{1+x^{2}}=\tan^{-1}\frac{2x}{1-x^{2}}$, is $-a< x< b$.
Find the value of $a+b$

If $\displaystyle \sin ^{-1}x-\sin ^{-1}y= \frac{\pi }{2}$ ,then

$\displaystyle \sin ^{-1}\left ( a-\frac{a^{2}}{3}+\frac{a^{3}}{9}\cdots \infty  \right )+\cos ^{-1}\left ( 1+b+b^{2}+b^{3}+\cdots \infty  \right )= \frac{\pi }{2}$ , when

The number of solutions of the equation  $2\sin^{-1}\left ( \sqrt{x^{2}-x+1} \right )+\cos^{-1}\left ( \sqrt{x^{2}-x} \right )= \displaystyle \frac{3\pi }{2}$ is

The equation $\sin ^{-1}x=2\sin ^{-1}a$ , has a solution for

Let $f(x) =e^{\displaystyle\cos^{-1}\sin \left ( x+\displaystyle\frac{\pi }{3} \right )}$, then $f\left (\displaystyle \frac{8\pi }{9} \right )$  equals

If $\displaystyle \cot ^{-1}\left [ \left ( \cos \alpha  \right )^{1/2} \right ]+\left [ \tan ^{-1}\left ( \cos \alpha  \right )^{1/2} \right ]=x$ , then $\sin x$ equals 

The domain of $\sin ^{-1}[x]$, where $[x]$ is greatest integer function, given by

If $\cos^{-1}x+\cos^{-1}y+\cos^{-1}z= 3\pi$, then value of $\displaystyle \sum xy$ equals