Inverse Trigonometric Functions

A value of $x$ satisfying the equation $\sin\,\left[\cot\,^{-1}\left(1+x\right)\right]=\cos\,\left[\tan\,^{-1}x\right],$ is:

If the inverse trigonometric functions take principal values, then…

Let $\left(a,b\right)\subset\,\left(0,2\pi\,\right)$be the largest interval for which $\sin\,^{-1}\left(\sin\,\theta\,\right)-\cos\,^{-1}\left(\sin\,\theta\,\right)>0,\theta\,\in\,\left(0,2\pi\,\right)$, holds . If…

The principal value of $\tan\,^{-1}\left(cot\frac{43\pi\,}{4}\right)$ is

For $k\in\,\mathbb{R}$, let the solutions of the equation $\cos\,\left(\sin\,^{-1}\left(x\cot\,\left(\tan\,^{-1}\left(\cos\,\left(\sin\,^{-1}x\right)\right)\right)\right)\right)=k,0<\left|x\right|<\frac{1}{\sqrt{2}}$be…

If $y=\underset{k=1}{\overset{6}{\sum\,}}k\cos\,^{-1}\left{\frac{3}{5}\cos\,kx-\frac{4}{5}\sin\,kx\right}$ then $\frac{dy}{dx}$at$x=0$is

Let$\tan\,^{-1}y=\tan\,^{-1}x+\tan\,^{-1}\left(\frac{2x}{1-x^{2}}\right)$, where $\left|x\right|<\frac{1}{\sqrt{3}}$,Then a value of$y$ is

The number of solutions of the equation $\sin\,^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos\,^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}$for $x\in\,[-1,1]$, and $[x]$ denotes the greatest integer less than or equal to $x,$is :

Let the maximum value of $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{\mathrm{m}}{\mathrm{n}} \pi^{2}$, where…

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_______.

$\tan\,^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right)+\sec\,^{-1}\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}=$

For $x\in\,(-1,1]$, the number of solutions of the equation $\sin\,^{-1}x=2\tan\,^{-1}x$ is equal to