$$ \sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right)= $$

$2 \cos ^{-1} x=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$ is valid for all values of ' $x$ ' satisfying

Let $f: R \rightarrow R$ be given $f(x)=\tan x$. Then, $f^{-1}(1)$ is

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$, then $x(y+z)+y(z+x)+z(x+y)$ equals to

If $2 \sin ^{-1} x-3 \cos ^{-1} x=4, x \in[-1,1]$, then $2 \sin ^{-1} x+3 \cos ^{-1} x$ is equal to

If $$\sin ^{-1}\left(\frac{2 a}{1+a^2}\right)+\cos ^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)$$ where $$a, x \in(0,1)$$, then the value of $$x$$ is

The value of $$\cot ^{-1}\left[\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right]$$, where $$x \in\left(0, \frac{\pi}{4}\right)$$ is

Domain $$\cos ^{-1}[x]$$ is, where [ ] denotes a greatest integer function

$$\cos \left[\cot ^{-1}(-\sqrt{3})+\frac{\pi}{6}\right]$$ is equal to

$$\tan ^{-1}\left[\frac{1}{\sqrt{3}} \sin \frac{5 \pi}{2}\right] \sin ^{-1}\left[\cos \left(\sin ^{-} \frac{\sqrt{3}}{2}\right)\right]$$ is equal to

If $$f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$$, then $$f^{\prime}(\sqrt{3})$$ is

The domain of the function defined by $$f(x)=\cos ^{-1} \sqrt{x-1}$$ is

The value of $$\cos \left(\sin ^{-1} \frac{\pi}{3}+\cos ^{-1} \frac{\pi}{3}\right)$$ is Does not exist

If $$f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$$ then $$f^{\prime}(0)=$$

$$\cos \left[2 \sin ^{-1} \frac{3}{4}+\cos ^{-1} \frac{3}{4}\right]=$$

If $$a+\frac{\pi}{2}<2 \tan ^{-1} x+3 \cot ^{-1} x< b$$ then '$$a$$' and '$$b$$' are respectively.