Chapter 1.7, Problem 30ES

The function ${\cot }^{-1}x$ is defined to be the inverse of the restricted cotangent function $\cot x,0<x<\pi$ and the function ${\csc }^{-1}x$ is defined to be the inverse of the restricted cosecant function $\csc x,-\pi /2\le x\le \pi /2,x\ne 0$ Use these definitions in these and in all subsequent exercises that involve these functions.

Show that

(a) ${\cot }^{-1}x=\left\{\begin{array}{cc}{\tan }^{-1}\left(1/x\right),& \text{if }x>0\\ \pi +{\tan }^{-1}\left(1/x\right),& \text{if }x<0\end{array}\right.$

(b) ${\sec }^{-1}x={\cos }^{-1}\frac{1}{x},\text{ if }\left|x\right|\ge 1$

(c) ${\csc }^{-1}x={\sin }^{-1}\frac{1}{x},\text{ if }\left|x\right|\ge 1$ .