Inverse Trigonometric Equations and Inequalities

The number of real solutions of ${\tan }^{-1}\sqrt{x\left(x+1\right)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$  is –

The value of $\sin \left({\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\frac{1}{2}\right)=?$

The $x$ satisfying ${\sin }^{-1}x+{\sin }^{-1}\left(1-x\right)={\cos }^{-1}x$ are

Consider the equation: ${\sin }^{–1}\left(\mathrm{x}–1\right)+{\cos }^{–1}\left(\mathrm{x}–3\right)+{\tan }^{–1}\left(\frac{x}{2–x^2}\right)=\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}k+\pi ,$ then the value of $k$ equals:

Considering only the principal values of inverse functions, which of the following holds true for the set $A=\left\{x\ge 0:{\tan }^{-1}\left(2x\right)+{\tan }^{-1}\left(3x\right)=\frac{\pi }{4}\right\}$

Evaluate $\cot \left(\sum _{n=1}^{19}{\cot }^{-1}\left(1+\sum _{p=1}^{n}2p\right)\right)$ :

If ${\sin }^{-1}\left(1-x\right)-2{\sin }^{-1}x=\frac{\pi }{2}$, then the value of $x$ is

Simplify: ${\sin }^{-1}\frac{4}{5}+2{\tan }^{-1}\frac{1}{3}.$

Find $x$, if $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c} x\right)$

If $f\left(x\right)={\sin }^{-1}\left(\frac{\sqrt{3}}{2}x-\frac{1}{2}\sqrt{1-x^2}\right), -\frac{1}{2}\le x\le 1,$ then $f\left(x\right)$equal to 

If ${\cos }^{-1}x>{\sin }^{-1}x$ then

If ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z=\mathrm{\pi }$, then the value of $x+y+z$ is _____

$\sum _{r=0}^{\infty }{\tan }^{-1}\left(\frac{1}{1+r+r^2}\right) =$

 If $3{\sin }^{-1}\frac{2x}{1+x^2}-4{\cos }^{-1}\frac{1-x^2}{1+x^2}+2{\tan }^{-1}\frac{2x}{1-x^2}=\frac{\mathrm{\pi }}{3}$, then the value of $x$ lying in $\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ is _____

If $x>1,$ then the value of $f\left(5\right)$, if $f\left(x\right)=2{\tan }^{-1}x+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right).$

Let a function $\mathrm{f}:(0,1)\to (0,1),\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)$ be defined implicitly as  ${\cot }^{-1}\left(\frac{1-x}{1+x}\right)+{\cot }^{-1}\left(\frac{1-f\left(x\right)}{1+f\left(x\right)}\right)=\frac{3\pi }{4}$ then

If $2^{{\left({\cos }^{-1}x\right)}^2}\sqrt{{\left({\sin }^{-1}y\right)}^4-2{\left({\sin }^{-1}y\right)}^2+2}=1$ then the value of $(x-y)$ can be

If ${\cot }^{-1}\left(\frac{n^2-10n+21·6}{\pi }\right)>\frac{\pi }{6},n\in N$ then $n$can be

Let $S_1$ is the complete solution set of the inequality $\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(x\right)>\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(x^2\right)$ and $S_2$ is the complete solution set of the inequality ${\left({\cot }^{-1}x\right)}^2-5\mathrm{c}\mathrm{o}{\mathrm{t}}^{-1}x+6>0$, then $S_1\cap S_2$ is

Let $p=2{\tan }^{-1}\left(\frac{1-x}{1+x}\right)$ and $q={\sin }^{-1}\left(\frac{2x}{1+x^2}\right).$ If $x<-1$, then the value of $\left(q-p\right)$ is