Inverse Trigono...

$$\sec^{-1}x+\tan^{-1}x$$ is real if $$\displaystyle x\epsilon \left ( -\infty ,-a \right ]\cup \left [ b,\infty  \right )$$.
Find the value of $$a+b$$.

If $$\cos^{-1}\lambda +\cos^{-1}\mu +\cos^{-1}\gamma =3\pi $$, then find the value of $$\lambda \mu +\mu \gamma +\gamma \lambda $$

Find the value of $$x$$ for which;  $${cosec}^{-1} \left( \cos x \right)$$ is real

Find the number of solutions of the equation $$\cos \left ( \cos^{-1}x \right )=cosec\:\left ( cosec^{-1}\:x \right )$$

Value of $$x$$ for which $$\displaystyle \cos^{-1}\left ( \frac{1-x^2}{1+x^2} \right )= 2\tan^{-1}x$$ satisfied is $$x\epsilon \left [ a,\infty  \right )$$.
Find the value of $$a.$$

If  the range for  $$y=\left ( \cot^{-1}x \right )\left ( \cot^{-1}\left ( -x \right ) \right )$$ is
$$0< y\leq \cfrac{\pi ^{a}}{b}$$.
Find the value of $$a+b$$

Evaluate the following:
i. $$\displaystyle \sin \left ( \cos^{-1}\frac{3}{5} \right )$$
ii. $$\displaystyle \cos \left ( \tan^{-1}\frac{3}{4} \right )$$
iii. $$\displaystyle \sin \left ( \frac{\pi }{2}-\sin^{-1}\left ( \frac{1}{2} \right ) \right )$$

Find the value of $$\displaystyle \sin^{-1}x+\sin^{-1}\frac{1}{x}+\cos^{-1}x+\cos^{-1}\frac{1}{x}$$

If the range of $$f\left ( x \right )=\sin^{-1}x+\tan^{-1}x+\cos^{-1}x$$ is
$$\displaystyle =\left [ \frac{\pi }{a}, \frac{b\pi }{c} \right ]$$,
Find the value of $$a+b+c$$

If $$\cos^{-1}x=\left\{\begin{matrix}
a\pi -b\cos^{-1}\left ( 2x^{2}-1 \right ), if\:-1\leq x< 0\\
c\cos^{-1}\left ( 2x^{2}-1 \right ), if\:0\leq x\leq  1
\end{matrix}\right.$$.
Find the value of $$a+b+c$$.