Inverse Trigono...

The value of $$ \lim_{\left | x \right |\rightarrow \infty} $$ $$ \cos \left ( \tan^{-1}\left ( \sin \left ( \tan^{-1} x \right ) \right ) \right ) $$ is equal to :

$$ \tan^{-1}a +\tan^{-1}b $$, where $$ a> 0, b> 0, ab> 1 $$ is equal to :

Let $$ f(x)=cosec^{-1}\:\left [ 1+\sin ^{2} x\right ] $$, where $$ \left [ . \right ]  $$ denotes the greatest integer function. Then $$ f(x) $$ equals;

If $$\displaystyle \tan \left ( \cos^{-1} x \right ) = \sin \left ( \cot^{-1} \frac{1}{2} \right )$$, then find the value of $$\displaystyle x$$

If $$\tan^{-1}\left ( \dfrac{x}{\sqrt{a^p-x^q}} \right )= \sin^{-1}\left ( \dfrac{x}{a} \right ),a> 0$$. Find the value of p and q.

If  $$\displaystyle \tan^{-1} \frac{1}{\sqrt{x^{a} - b}} = \frac{\pi}{c} - \sec^{-1} x$$, $$\displaystyle \left | x \right | > 1$$.
Find the value of a,b and c.

The value of $$\displaystyle \sin ^{-1}\left ( \sin 2 \right )$$ is?

Find the minimum value the function $$\displaystyle f\left ( x \right )=\frac{\pi ^{2}}{16\cot^{-1}\left ( -x \right )}-\cot^{-1}x$$

For  $$\displaystyle \tan^{-1}\left ( \frac{1-x}{1+x} \right )$$, $$0\leq x\leq 1$$.
What is the sum of the smallest and the largest values of function.

Find the number of real solutions :
$$ \tan^{-1}\sqrt{x\left ( x+1 \right )}+\sin^{-1}\sqrt{x^2+x+1}= \dfrac{\pi }{2} $$