Properties of Inverse Trigonometric Functions

Let $y={\sin }^{-1}\left(\sin 8\right)-{\tan }^{-1}\left(\tan 10\right)+{\cos }^{-1}\left(\cos 12\right)$$-{\sec }^{-1}\left(\sec 9\right)+{\cot }^{-1}\left(\cot 6\right)-{\mathrm{cosec}}^{-1}\left(\mathrm{cosec}7\right)$. If $y$ simplifies to $a\pi +b$, then $\left(a-b\right)$ is

If ${\sin }^{-1}\left(\mathrm{x}-\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{4}-\dots \dots .\right)+{\cos }^{-1}\left({\mathrm{x}}^2-\frac{{\mathrm{x}}^4}{2}+\frac{{\mathrm{x}}^6}{4}-\dots \dots .\right)=\frac{\mathrm{\pi }}{2}$ for $0<\left|\mathrm{x}\right|<\sqrt{2}$ then $\mathrm{x}$ equals to

The principal value of ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is:

Which of the following is the value of given expression : $\frac{9\pi }{8}-\frac{9}{4}{\sin }^{-1}\left(\frac{1}{3}\right)$

Evaluate ${\mathrm{cosec}}^{-1}\left(2\cos \frac{5\mathrm{\pi }}{4}\right)$

Evaluate ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{5\mathrm{\pi }}{6}\right)$

Evaluate ${\mathrm{cosec}}^{-1}\left(2\sin \frac{3\mathrm{\pi }}{4}\right)$

Evaluate ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{3\mathrm{\pi }}{4}\right)$

Evaluate ${\mathrm{cosec}}^{-1}\left(\mathrm{cosec}\frac{25\mathrm{\pi }}{6}\right)$

Using the identity ${\sin }^{-1}\left(\sin x\right)=x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}.$

Statement I: ${\sin }^{-1}\left(\sin 2\right)=2$

Statement II: The principal value of ${\sin }^{-1}\left(\sin x\right)=x$

The number of real solutions of $\left(x, y\right)$ where $y=\left|\sin x\right|,$ $y={\cos }^{-1}\left(\cos x\right),$ $-2\pi \le x\le 2\pi ,$ is

If $a+\frac{\pi }{2}<2 {\tan }^{-1}x+3 {\cot }^{-1}x<b$ then $\text{'}a\text{'}$ and $\text{'}b\text{'}$ are respectively.

The value of $x$ for which $\sin \left({\cot }^{-1}\left(1+x\right)\right)=\cos \left({\tan }^{-1}x\right)$ is

$\underset{-1}{\overset{3}{\int }}\left[ta{n}^{-1}\frac{x}{x^2+1}+ta{n}^{-1}\frac{x^2+1}{x}\right]dx=$

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

Evaluate $\text{cos}\left[{\text{cos}}^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi }{6}\right]$

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

If ${\sin }^{-1}\left(\sin x\right)=\pi -x,$ then $x$ belongs to $\left[\frac{\mathrm{a\pi }}{b},\frac{c\mathrm{\pi }}{d}\right]$ then the value of $a+b+c+d$ is equal to 

Number of solutions for ${\tan }^{-1}\left(\frac{x+1}{x-1}\right)+{\tan }^{-1}\left(\frac{x-1}{x}\right)={\tan }^{-1}\left(-7\right)$ is