Chapter 5.1, Problem 40E

To determine

To simplify: The given trigonometric expression and check the result using the table feature of a graphing utility.

Answer to Problem 40E

  $\frac{1+{\tan }^2\theta }{{\sec }^2\theta }=1$

The result is numerically verified.

Explanation of Solution

Given information:

The given trigonometric expression:

  $\frac{1+{\tan }^2\theta }{{\sec }^2\theta }$

Formula used:

Pythagorean identity:

  ${\sec }^2\theta =1+{\tan }^2\theta$

Calculation:

As per problem,

The given trigonometric expression

  $\frac{1+{\tan }^2\theta }{{\sec }^2\theta }$

Use the Pythagorean identity: ${\sec }^2\theta =1+{\tan }^2\theta$ , to write the expression in terms of secant.

  $\begin{array}{l}\frac{1+{\tan }^2\theta }{{\sec }^2\theta }=\frac{{\sec }^2\theta }{{\sec }^2\theta }\text{ [Pythagorean identity]}\\ \text{ = 1 [Divide out common factors and simplify]}\\ \end{array}$

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions $\text{Y}=\frac{1+{\tan }^2\theta }{{\sec }^2\theta }$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

From the graph, $\text{Y}=1$ for all points along x-axis.

The table of values shows points (indicated in black in the graph) for the expression.

  

Interpretation: According to the above graph, it can be observed that $\text{Y}=1$ .

Therefore, from the graph, $\frac{1+{\tan }^2\theta }{{\sec }^2\theta }=1$ , which is same as calculated. Thus, the result is verified.