Inverse Trigono...

The largest interval lying in $$\left ( \dfrac{-\pi }{2},\dfrac{\pi }{2} \right )$$ for which the function $$\left [ f(x)=4^{-x^{2}}+\cos^{-1}\left ( \dfrac{x}{2}-1 \right )+\log (\cos x) \right ]$$ is defined, is-

Statement I :  The equation $$(sin^{-1}x)^3+(cos^{-1}x)^3-a\pi^3=0$$ has a solution for all $$a\geqslant \dfrac {1}{32}.$$
Statement II : For any $$x\epsilon R, sin^{-1}x+cos^{-1}x=\dfrac {\pi}{2}$$ and $$0\leq (sin^{-1}x-\dfrac {\pi}{4})^2\leq \dfrac {9\pi^2}{16}$$.

$${cos}^{-1}\left \{ \frac{1}{2}x^2+\sqrt{1-x^2}.\sqrt{1-\frac{x^2}{4}} \right \}={cos}^{-1}\frac{x}{2}-{cos}^{-1}x$$ holds for

Solve: $$\displaystyle { \tan }^{ -1 }\left( \frac { x }{ y }  \right) -{ \tan }^{ -1 }\frac { x-y }{ x+y } $$ is equal to

Value of $$\tan^{-1}\begin{Bmatrix}\dfrac{\sin\,2-1}{\cos\,2}\end{Bmatrix}$$ is

The number of triplets $$\left ( x,y,z \right )$$ satisfies the equation $$\displaystyle f\left ( x,y,z \right )= \sin ^{-1}x+\sin ^{-1}y+\sin ^{-1}z= \dfrac{3\pi }{2}$$ is

Calculate the value of   $$\displaystyle \sin^{-1} \cos \left ( \sin^{-1} x\right )  + \cos^{-1} \sin \left ( \cos^{-1} x \right ) $$. where $$\displaystyle\left | x \right |  \leq  1$$

Sin-1 $$(\displaystyle \frac{3}{5})+\mathrm{S}\mathrm{i}\mathrm{n}^{-1}(\frac{5}{13})=\sin_{X}^{-1}$$ then$$\mathrm{x}=$$

If $$\sec^{-1} x+ \sec^{-1}y + \sec^{-1}z = 3\pi$$, then $$xy + yz + zx =$$ _______.

Find value of $$\cos^{-1}\left(-\dfrac {1}{2}\right)$$.