Chapter 5.5, Problem 46E

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 46E

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

  $\frac{1}{4}\left(\frac{3-4\cos 2x+\cos 4x}{1+\cos 2x}\right)$

Explanation of Solution

Given information:

The following expression:

  ${\sin }^{2}x{\tan }^{2}x$

Formula used:

The following power reduction formula

  $\begin{array}{l}{\sin }^{2}u=\frac{1-\cos 2u}{2}\\ {\cos }^{2}u=\frac{1+\cos 2u}{2}\\ {\tan }^{2}u=\frac{1-\cos 2u}{1+\cos 2u}\end{array}$

The following expansion:

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

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

  $\begin{array}{l}{\sin }^2{\tan }^{2}x=\left(\frac{1-\cos 2x}{2}\right)\left(\frac{1-\cos 2x}{1+\cos 2x}\right)\text{ [Power reduction formula]}\\ =\frac{{\left(1-\cos 2x\right)}^2}{2(1+\cos 2x)}\text{ [Multiply]}\\ =\frac{1+{\cos }^{2}2x-2\cos 2x}{2(1+\cos 2x)}\text{ [Find square]}\\ =\frac{1+\frac{1+\cos 4x}{2}-2\cos 2x}{2(1+\cos 2x)}\text{ [Power reduction formula]}\\ =\frac{1+\frac{1}{2}+\frac{1}{2}\cos 4x-2\cos 2x}{2(1+\cos 2x)}\text{ [Seperate fractions]}\\ =\frac{\frac{1}{2}(2+1+\cos 4x-4\cos 2x)}{2(1+\cos 2x)}\text{ [Factor]}\\ =\frac{\frac{1}{2}(3+\cos 4x-4\cos 2x)}{2(1+\cos 2x)}\text{ [Add like terms]}\\ =\frac{1}{2}(3+\cos 4x-4\cos 2x)\cdot \frac{1}{2(1+\cos 2x)}\text{ [Multiply by reciprocal of denominator]}\\ =\frac{1}{4}\cdot \frac{3-4\cos 2x+\cos 4x}{1+\cos 2x}\text{ [Simplify]}\end{array}$

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 ${\text{Y}}_{\text{1}}={\sin }^{2}x{\tan }^{2}x$ and ${\text{Y}}_2=\frac{1}{4}\cdot \left(\frac{3-4\cos 2x+\cos 4x}{1+\cos 2x}\right)$ 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