Chapter 5.1, Problem 34E

To determine

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

Answer to Problem 34E

  $\cos x(1+{\tan }^{2}x)=\sec x$

The result is numerically verified.

Explanation of Solution

Given information:

The given trigonometric expression:

  $\cos x(1+{\tan }^{2}x)$

Formula used:

Pythagorean identity:

  ${\sec }^{2}x=1+{\tan }^{2}x$

Reciprocal identity:

  $\sec x=\frac{1}{\cos x}$

Calculation:

As per problem,

The given trigonometric expression:

  $\cos x(1+{\tan }^{2}x)$

Use the Pythagorean identity ${\sec }^{2}x=1+{\tan }^{2}x$ to simplify.

  $\begin{array}{l}\cos x(1+{\tan }^{2}x)=\cos x({\sec }^{2}x)\text{ [Pythagorean identity]}\\ \text{ = }\cos x\left(\frac{1}{{\cos }^{2}x}\right)\text{ [Reciprocal identity]}\\ \text{ =}\frac{1}{\cos x}\text{ [Divide out common factors]}\\ \text{ = }\sec x\text{ [Reciprocalidentity]}\end{array}$

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 ${\text{Y}}_1=\cos x(1+{\tan }^{2}x)$ and ${\text{Y}}_2=\sec x$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

The curve of ${\text{Y}}_1$ is indicated by yellow line and the curve of ${\text{Y}}_2$ is indicated by black line. Both the curves coincide.

The table of values shows points (indicated in black in the graph) that coincide for both the expressions.

  

Interpretation: According to the above graph, it can be observed that ${\text{Y}}_1={\text{Y}}_2$ as both the graphs coincide.

Therefore numerically, $\cos x(1+{\tan }^{2}x)=\sec x$ .