Chapter 1.6, Problem 46E

(a)

To determine

To show: Thefunction $\csc x$ is an odd function of $x$ .

Explanation of Solution

Given information:The given function is $\csc x$ .

Proof:

Simplify the given function

  $\csc x=\frac{1}{\sin x}$

The $\sin x$ is an odd function of $x$ . So,

  $\begin{array}{c}\csc \left(-x\right)=\frac{1}{\sin \left(-x\right)}\\ =-\frac{1}{\sin x}\\ =-\csc x\end{array}$

As $\csc \left(-x\right)=-\csc x$ , the cotangent is an odd function.

Hence, it is proved that $\csc x$ is an odd function of $x$ .

(b)

To determine

To show: The reciprocal of a function is an odd function.

Explanation of Solution

Proof:

A function $f$ is said to be odd function if $f\left(-x\right)=-f\left(x\right)$ for all $x$ in its domain

Assume that $f\left(x\right)$ is an odd function. Substitute $-x$ for $x$ in the function.

  $\begin{array}{c}\frac{1}{f\left(-x\right)}=\frac{1}{-f\left(x\right)}\\ =-\frac{1}{f\left(x\right)}\end{array}$

So, the reciprocal of a function $\frac{1}{f}$ is odd.

Hence, it is proved that the reciprocal of a function is an odd function.