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