Chapter 0.6, Problem 46E

i.

To determine

To find: Show that $\csc x$ is an odd function of $x$ .

Answer to Problem 46E

Hence, the $\csc x$ function is an odd function of $x$ .

Explanation of Solution

Given: A function is $\csc x$ .

Concept Used:

If $f(-x)=f(x)$ for all values of $x$ then function is called even and if $f(-x)=-f(x)$ for all values of $x$ then function is called odd.

Calculation:

  $f(x)=\text{csc }x$

Now,

  $\begin{array}{c}f(-x)=\text{csc}(-x)\\ =-\text{csc }x\left\{\because \text{csc}\left(-\theta \right)=-\text{csc}\left(\theta \right)\right\}\\ =-f(x)\end{array}$

Therefore, the $\csc x$ function is an odd.

Conclusion:

Hence, the $\csc x$ function is an odd function of $x$ .

ii.

To determine

To find: Show that the reciprocal of an odd function is odd.

Answer to Problem 46E

Hence, the reciprocal of an odd function is odd.

Explanation of Solution

Given: An odd function

Concept Used:

If $f(-x)=f(x)$ for all values of $x$ then function is called even and if $f(-x)=-f(x)$ for all values of $x$ then function is called odd.

Trigonometric identity: $\csc \theta =\frac{1}{\sin \theta }$

Calculation:

Let the odd function be $f\left(x\right)$

Hence, $f\left(-x\right)=-f\left(x\right)$

And reciprocal of $f\left(x\right)$ is $g\left(x\right)=\frac{1}{f\left(x\right)}$

Now, substitute $-x$ for $x$ in $g\left(x\right)$:

Since,

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

  $g\left(-x\right)=-g\left(x\right)$

Thus, the reciprocal of the function is also odd function.

Conclusion:

Hence, the reciprocal of an odd function is odd.