G Tewani Solutions for Chapter: Inverse Trigonometric Functions, Exercise 3: DPP

If $x$ and $y$ are positive integers satisfying ${\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}\left(\frac{1}{y}\right)={\tan }^{-1}\left(\frac{1}{7}\right),$ then the number of ordered pairs of $\left(x, y\right)$ is

If $x\in \left[-1, \frac{-1}{\sqrt{2}}\right],$ then the inverse of the function $f\left(x\right)={\sin }^{-1}\left(2x\sqrt{1-x^2}\right)$ is given by

The expression $\sum _{n=1}^{\infty }{\cot }^{-1}\left(n^2-3n+3\right)$ simplifies to

The value of ${\displaystyle \sum _{n=1}^{\infty }}{\cot }^{-1}\left(\frac{\left(n^2+2n\right)\left(n^2+2n+1\right)+1}{2n+2}\right)$ is equal to

A solution of the equation $2{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)-\pi x^3=0$ is

The solutions of the equation $\sin \left[2{\cos }^{-1}\left\{\cot \left(2{\tan }^{-1}x\right)\right\}\right]=0$ are

Let $x_1, x_2, x_3, x_4$ be four non-zero numbers satisfying the equation ${\tan }^{-1}\frac{\alpha }{x}+{\tan }^{-1}\frac{b}{x}+{\tan }^{-1}\frac{c}{x}+{\tan }^{-1}\frac{d}{x}=\frac{\pi }{2}$ . Then

Which of the following is/are true?