Embibe Experts Solutions for Chapter: Inverse Trigonometric Functions, Exercise 1: Exercise 1

If $x^2+y^2+z^2=r^2$ and $x, y, z>0$, then ${\tan }^{-1}\left(\frac{xy}{zr}\right)+{\tan }^{-1}\left(\frac{yz}{xr}\right)+{\tan }^{-1}\left(\frac{zx}{yr}\right)$ is equal to

If $a<\frac{1}{32}$, then the number of solution of ${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3=a{\pi }^3$ is

The expression $\tan \left(\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}x\right)+\tan \left(\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}x\right)$ equals -

If $k$ satisfies $k^2-k-6>0$, then a value exists for -

The domain and range of $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x+{\cot }^{-1}x+{\sec }^{-1}x+{cosec}^{-1}x$  respectively are -

Let $f\left(x\right)={\sec }^{-1}\left(x-10\right)+{\cos }^{-1}\left(10-x\right)$. The range of $f\left(x\right)$ is

${\sin }^{-1}\left(-\left(\frac{1}{2}\right)\right)+{\cos }^{-1}\left(-\left(\frac{1}{2}\right)\right)+{\cot }^{-1}\left(-\sqrt{3}\right)+{cosec}^{-1}\left(\sqrt{2}\right)+{\tan }^{-1}\left(-1\right)+{\sec }^{-1}\left(\sqrt{2}\right)$ equals

If $x_1={\cos }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$ and $x_2={\sin }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then