Inverse Trigono...

If $$\displaystyle \sin \:^{-1}\left ( x-\frac{x^{}}{2}+\frac{x^{3}}{4}-... \right )+\cos ^{-1}\left ( x^{2}-\frac{x^{4}}{2}+\frac{x^{6}}{4}-... \right )=\frac{\pi }{2}$$ for $$\displaystyle  0<\left | x \right |<\sqrt{2}$$, then $$x$$ equals

The value of $$\displaystyle sin^{-1}(cos(cos^{-1}(cos\:x)+sin^{-1}(sin\:x)))$$, where $$\displaystyle x\:\epsilon\:\left ( \frac{\pi}{2},\pi \right )$$, is equal to 

$$\displaystyle \:\sin \cot ^{-1}\tan \cos ^{-1}x$$ is equal to

 If $$\displaystyle \frac{1}{\sqrt{2}}< x< 1$$ then $$\displaystyle\cos ^{-1}x+\cos ^{-1}\left ( \frac{x+\sqrt{1-x^{2}}}{\sqrt{2}} \right )$$ is equal to 

If $$\displaystyle \cos \:^{-1}x+\cos \:^{-1}y+\cos ^{-1}z=3\pi$$ then the value of $$xy+yz+zx$$ is

$$\displaystyle sec^{2}(tan^{-1}2)+cosec^{2}(cot^{-1}3)$$ is equal to 

If $$\displaystyle f\left ( x \right ) = \sin^{-1}x + \sec^{-1} x$$ is defined, then which of the following value/values is/are in its range?

The value of $$\displaystyle sin^{-1}\left[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^{2}}\right]$$ is equal to

The number of integer $$x$$ satisfying $$\displaystyle sin^{-1}|x-2|+cos^{-1}(1-|3-x|)=\frac{\pi}{2}$$ is

If $$x_{ 1 }=2\: tan^{ -1 }\left( \frac { 1+x }{ 1-x }  \right) ,x_{ 2 }=sin^{ -1 }\left( \frac { 1-x^{ 2 } }{ 1+x^{ 2 } }  \right) $$, where $$x_1,x_2\:\epsilon\:(0,1)$$, then $$2\left( x_{ 1 }+x_{ 2 } \right) $$ is equal  to