Chapter 5, Problem 100CR

To determine

To evaluate: the expression in terms of the first power of the cosine and verify it using graphing utility.

Answer to Problem 100CR

  ${\sin }^{2}2x{\tan }^{2}2x=\frac{3-4\cos 4x+\cos 8x}{4+4\cos 4x}$

Explanation of Solution

Given:

  ${\sin }^{2}2x{\tan }^{2}2x$

The following identities are being used here,

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

So,

  $\begin{array}{c}{\sin }^{2}2x{\tan }^{2}2x=\left(\frac{1-\cos 4x}{2}\right)\left(\frac{1-\cos 4x}{1+\cos 4x}\right)\\ =\frac{{\left(1-\cos 4x\right)}^2}{2\left(1+\cos 4x\right)}\\ =\frac{1-2\cos 4x+{\cos }^{2}4x}{2+2\cos 4x}\end{array}$

Again substituting,

  ${\cos }^{2}4x=\frac{1+\cos 8x}{2}$

So,

  $\begin{array}{c}{\sin }^{2}2x{\tan }^{2}2x=\frac{1-2\cos 4x+\frac{1+\cos 8x}{2}}{2+2\cos 4x}\\ =\frac{2-4\cos 4x+1+\cos 8x}{4+4\cos 4x}\\ =\frac{3-4\cos 4x+\cos 8x}{4+4\cos 4x}\end{array}$

So, the first power is derived in the above equation.

Now verifying with the help of graph,

The graph shown below is of ${\sin }^{2}2x{\tan }^{2}2x$

  

Now, the graph below is of $\frac{3-4\cos 4x+\cos 8x}{4+4\cos 4x}$

  

Hence it is verified that above graphs are similar.