Chapter 5.1, Problem 49E

To determine

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

Answer to Problem 49E

  ${\tan }^{4}x+2{\tan }^{2}x+1={\sec }^{4}x$

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:

  ${\tan }^{4}x+2{\tan }^{2}x+1$

Formula used:

The square of a sum can be written as:

  ${(a+b)}^2=a^2+b^2+2ab$

Calculation:

As per problem,

The given trigonometric expression

  ${\tan }^{4}x+2{\tan }^{2}x+1$

The expression has the polynomial form $a^2+b^2+2ab$ and can be factorized as:

  $\begin{array}{l}{\tan }^{4}x+2{\tan }^{2}x+1={(}^{{\tan }^2}+2{\tan }^{2}x+1\text{ [Rule of exponents]}\\ \text{ = }{(}^{{\tan }^2}+2\cdot {\tan }^{2}x\cdot 1+{(}^1\text{ [Write in }{a}^2+b^2+2ab\text{ form]}\\ \text{ = (}{\tan }^{2}x+1{)}^2\text{ [Use (}a+b{)}^2\text{ formula]}\\ {\text{ = (sec}}^{2}x{)}^2\text{ [Pythagorean identity: }{\tan }^{2}x+1={\text{sec}}^{2}x\text{ ]}\\ {\text{ = sec}}^{4}x\text{ [ Rule of exponents]}\end{array}$

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 ${\text{Y}}_1={\tan }^{4}x+2{\tan }^{2}x+1$ 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 red line. Both the curves coincide.

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

Therefore, from the graph, ${\tan }^{4}x+2{\tan }^{2}x+1={\sec }^{4}x$ , which is same as calculated. Thus, the result is graphically verified.