Inverse Trigonometric Functions

If $\(\sin$ $\left(\cot$ $^{-1}(1+\mathrm{x})\right)=\cos$ $\left(\tan$ $^{-1}$ $\mathrm{x}\right)\),$ then $\(\mathrm{x}=\)$

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

If in a $\triangle A B C, \angle C=\frac{\pi}{2}$, then $\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)=$

$\cot ^{-1}(21)+\cot ^{-1}(13)+\cot ^{-1}(-8)=$

$\sin ^{-1}\left[x \sqrt{1-x}-\sqrt{x} \sqrt{1-x^{2}}\right]=$

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta-8$ and $\sec ^2\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^2\left(\cot ^{-1} \beta\right)-36$, then $\alpha^2+\beta$ is_______.

What is the domain of the function $f(x)=\cos ^{-1}(x-2)$ ?

$\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}$ holds, when

If $\cos ^{-1} x=\sin ^{-1} x$, then which one of the following is correct?

What is $\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{3}{5}\right)$ equal to?

The principal value of $\sin ^{-1} x$ lies in the interval

What is the principal value of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ ?