Chapter 5.5, Problem 42E

To determine

To write: The given expression in terms of the first power of cosine and verify the answer using a graphing utility.

Answer to Problem 42E

The given expression in terms of the first power of cosine can be written as follows:

  $\frac{1}{2}(1+\cos 4x)$

Explanation of Solution

Given information:

The following expression:

  ${\cos }^{2}2x$

Formula used:

The following power reduction formula

  ${\cos }^{2}u=\frac{1+\cos 2u}{2}$

Calculation:

The given expression can be written in terms of the first power of cosine as follows:

  $\begin{array}{l}{\cos }^{2}2x=\frac{1+\cos 2(2x)}{2}\text{ [Power reducing formula]}\\ =\frac{1+\cos 4x}{2}\text{ [Multiply]}\\ =\frac{1}{2}+\frac{1}{2}\cos 4x\text{ [Seperate the fractions]}\\ =\frac{1}{2}(1+\cos 4x)\text{ [Factor]}\end{array}$

The solution is verified graphically as follows:

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 ${\text{Y}}_{\text{1}}={\cos }^{2}2x$ and ${\text{Y}}_2=\frac{1}{2}(1+\cos 4x)$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation :

Since both the curves coincide, the solution is verified