Amit M Agarwal Solutions for Chapter: Inverse Trigonometric Functions, Exercise 2: Target Exercises

The number of the real solution of  $\sqrt{1+\cos 2x}=\sqrt{2}{\sin }^{-1}\left(\sin x\right), -\pi \le x\le \pi$ is

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

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

The set of values of $k$ for which $x^2-kx+{\sin }^{-1}\left(\sin 4\right)>0$ for all real $x$, is

The sum of the infinite series ${\cot }^{-1}\left(2\right)+{\cot }^{-1}8+{\cot }^{-1}18+{\cot }^{-1}32+\dots$ is 

If $x_1, x_2, x_3$ and $x_4$ are roots of the equation $x^4-x^3\sin 2\beta +x^2\cos 2\beta -x\cos \beta -\sin \beta =0$, then $\sum _{i=1}^4{\tan }^{-1}{x}_i$ is equal to

If $\sum _{i=1}^{2n}{\cos }^{-1}{x}_i=0$, then $\sum _{i=1}^{2n}{x}_i$ is

If $\sum _{i=1}^{20}{\sin }^{-1}{x}_i=10\pi$, then $\sum _{i=1}^{20}{x}_i$ is equal to