Properties of Inverse Trigonometric Functions

${\tan }^{-1}\left(\frac{x}{y}\right)-{\tan }^{-1}\frac{x-y}{x+y}$ is equal to

If ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\frac{\pi }{2},$ then $x$ is equal to

$\sin \left({\tan }^{-1}x\right), \left|x\right|<1$ is equal to

Solve $\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\frac{1-x}{1+x}=\frac{1}{2}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}x,\left(x>0\right)$

Solve $2 {\tan }^{-1}(\cos x)={\tan }^{-1}(2 cosec x)$

Prove that $\frac{9\pi }{8}-\frac{9}{4}{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\frac{1}{3}=\frac{9}{4}{\sin }^{-1}\frac{2\sqrt{2}}{3}$

Prove that, $\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)=\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}x, -\frac{1}{\sqrt{2}}\le x\le 1$

Prove that, ${cot}^{-1}\left(\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}}\right)=\frac{x}{2},x\in \left(0,\frac{\pi }{4}\right)$

Prove that, $ta{n}^{-1}\sqrt{x}=\frac{1}{2}co{s}^{-1}\left(\frac{1-x}{1+x}\right),x\in \left[0,1\right]$

Prove that ${\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{8}=\frac{\pi }{4}$

Prove that ${\tan }^{-1}\frac{63}{16}={\sin }^{-1}\frac{5}{13}+{\cos }^{-1}\frac{3}{5}$

Prove that ${\cos }^{-1}\frac{12}{13}+{\sin }^{-1}\frac{3}{5}={\sin }^{-1}\frac{56}{65}$

Prove that ${\cos }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{12}{13}={\cos }^{-1}\frac{33}{65}$

Prove that : ${\sin }^{-1}\frac{8}{17}+{\sin }^{-1}\frac{3}{5}={\tan }^{-1}\frac{77}{36}$

Prove that : $2{\sin }^{-1}\frac{3}{5}={\tan }^{-1}\frac{24}{7}$

Find the value of $\tan \left({\sin }^{-1}\frac{3}{5}+{\cot }^{-1}\frac{3}{2}\right)$

Find the value of: ${\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$

Find the value of ${sin}^{-1}\left(sin\frac{2\pi }{3}\right)$

If ${tan}^{-1}\frac{x-1}{x-2}+{tan}^{-1}\frac{x+1}{x+2}=\frac{\pi }{4}$, then find the value of $x$?

If $\mathrm{s}\mathrm{i}\mathrm{n}\left({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x\right)=1$, then find the value of $x$?