Chapter 5.1, Problem 46E

To determine

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

Answer to Problem 46E

  ${\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x={\sec }^{4}x$

The result is graphically verified as the curve of the given expression coincides at all points with the graph of its result.

Explanation of Solution

Given information:

The given trigonometric expression:

  ${\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x$

Formula used:

The following Pythagorean identity:

  ${\sec }^{2}x={\tan }^{2}x+1$

Calculation:

As per problem,

The given trigonometric expression

  ${\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x$

First factor out a common monomial factor and then use the fundamental identity.

  $\begin{array}{l}{\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x={\sec }^{2}x({\tan }^{2}x+1)\text{ [Factor out common monomial factor]}\\ {\text{ = sec}}^{2}x\cdot {\sec }^{2}x\text{ [Pythagorean identity]}\\ {\text{ = sec}}^{4}x\text{ [Multiply]}\\ \end{array}$

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 ${\text{Y}}_1={\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x$ and ${\text{Y}}_2={\sec }^{4}x$ , 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 shows some of the points (indicated in black in the graph) that coincide for both the expressions.

  

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

Therefore, from the graph, ${\sec }^{2}x{\tan }^{2}x+{\sec }^{2}x={\sec }^{4}x$ , which is same as calculated. Thus, the result is graphically verified.