Properties of Inverse Trigonometric Functions

If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\pi }{2}$, then the value of $x^{100}+y^{100}+z^{100}-\frac{3}{x^{101}+y^{101}+z^{101}}$ is

The value of ${\cot }^{-1}\left\{\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right\}$ is $\left(0<x<\frac{\pi }{2}\right)$

If ${\sin }^{-1}\frac{1}{x}={\sin }^{-1}\frac{1}{a}+{\sin }^{-1}\frac{1}{b}$, then the value of $x$ is

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{3}$, then find the value of ${\cos }^{-1}x+{\cos }^{-1}y$

Find the value of ${\cos }^{-1}\cos \left(\frac{7\pi }{6}\right)$

If $\alpha ={\sin }^{-1}\frac{\sqrt{3}}{2}+{\sin }^{-1}\frac{1}{3}$ and $\beta ={\cos }^{-1}\frac{\sqrt{3}}{2}+{\cos }^{-1}\frac{1}{3}$, then 

Find the value of $x$ and $y$ 

${\sin }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{3}, {\cos }^{-1}x-{\cos }^{-1}y=\frac{\pi }{3}$

The value of ${\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right)+{\cot }^{-1}\left(3\right)$ is

${\cos }^{-1}x$ is equal to 

Find the value of ${\sin }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{12}{13}+{\sin }^{-1}\frac{16}{65}$

Find the value of $\sin \left({\cot }^{-1}x\right)$.

Find the value of ${\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3}={\sin }^{-1}\frac{1}{\sqrt{5}}+{\cot }^{-1}3$

Find the value of ${\tan }^{-1}\sqrt{\left\{\frac{a\left(a+b+c\right)}{bc}\right\}}+{\tan }^{-1}\sqrt{\left\{\frac{b\left(a+b+c\right)}{ac}\right\}} +{\tan }^{-1}\sqrt{\left\{\frac{\mathrm{c}\left(a+b+c\right)}{\mathrm{ab}}\right\}}$

${\tan }^{-1}\left[\cos \left\{2{\tan }^{-1}\left(\frac{3}{4}\right)\right\}+\sin \left\{2{\cot }^{-1}\left(\frac{1}{2}\right)\right\}\right]$ is 

The number of solution of the equation ${\tan }^{-1}\left(x-1\right)+{\tan }^{-1}x+{\tan }^{-1}\left(x+1\right)={\tan }^{-1}3x$ is

The value of $\sin \left(4{\tan }^{-1}\frac{1}{3}\right)-\cos \left(2{\tan }^{-1}\frac{1}{7}\right)$ is

If ${\cos }^{-1}\frac{p}{a}+{\cos }^{-1}\frac{q}{b}=\alpha$, then $\frac{p^{\mathit{2}}}{a^{\mathit{2}}}-\frac{2pq}{ab}\cos \alpha +\frac{q^2}{ b^2}$ is equal to

The value of $2{\tan }^{-1}\left(cosec{\tan }^{-1}x-{\mathrm{tancot}}^{-1}x\right), x>0$ is

If ${\left({\tan }^{-1}x\right)}^2+{\left({\cot }^{-1}x\right)}^2=\frac{29{\pi }^2}{36}$, then $x$ equals

Find the value of ${\cos }^{-1}x+{\cos }^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3x^2}}{2}\right)$ where $\left(\frac{1}{2}\le x\le 1\right)$.