Chapter 5, Problem 17RE

To determine

To simplify: the expression, check result numerically.

Answer to Problem 17RE

  $\frac{\tan \left(-\theta \right)}{\sec \theta }=\left(-\sin \theta \right)$

Explanation of Solution

Given information:

Given expression

  $\frac{\tan \left(-\theta \right)}{\sec \theta }$ .

Calculation:

Let us consider the following expression, $\frac{\tan \left(-\theta \right)}{\sec \theta }$

Now, simplify the expression as shown:

  $\begin{array}{c}\frac{\tan \left(-\theta \right)}{\sec \theta }=-\frac{\tan \theta }{\frac{1}{\cos \theta }}\\ =-\frac{\frac{\sin \theta }{\cos \theta }}{\frac{1}{\cos \theta }}\\ =-\sin \theta \end{array}$

Now, check the result graphically as shown:

Table of values for $\frac{\tan \left(-\theta \right)}{\sec \theta }$ is shown:

  

Graph of $\frac{\tan \left(-\theta \right)}{\sec \theta }$ is constructed as

  

Table of values for $-\sin \theta$ is shown:

  

Graph of $-\sin \theta$

  

Both graphs are same and the functional values are also same.