Chapter 13.8, Problem 43WE

To determine

To Express: The $\cos 4\theta$ in terms of only of $\cos \theta$.

Answer to Problem 43WE

The $\cos 4\theta$ in terms of $\cos \theta$ alone is $8{\cos }^4\theta -8{\cos }^2\theta +1$ .

Explanation of Solution

Given information:

The given information is $\cos 4\theta$ .

Calculation:

Use the double angle formula for cosine and rewrite the given expression.

  $\begin{array}{l}\cos 4\theta ={\cos }^2\theta -{\sin }^2\theta \\ =\cos 2\theta \cos 2\theta -\sin 2\theta \sin 2\theta \end{array}$

Now, use the double angle formula for cosine and sine functions.

  $\cos 2\theta \cos 2\theta -\sin 2\theta \sin 2\theta ={\left(2{\cos }^2\theta -1\right)}^2{\left(2\sin \theta \cos \theta \right)}^2$

Expand the square terms and simplify.

  $\begin{array}{l}{\left(2{\cos }^2\theta -1\right)}^2{\left(2\sin \theta \cos \theta \right)}^2=4{\cos }^4\theta -4{\cos }^2\theta +1-4{\sin }^2\theta {\cos }^2\theta \\ =4{\cos }^4\theta -4{\sin }^2\theta {\cos }^2\theta -4{\cos }^2\theta +1\\ =4{\cos }^4\theta -4{\cos }^2\theta \left({\sin }^2\theta +1\right)+1\end{array}$

We know that

  ${\sin }^2\theta =1-{\cos }^2\theta$

Now,

  $\begin{array}{l}4{\cos }^4\theta -4{\cos }^2\theta \left(1-{\cos }^2\theta +1\right)+1=4{\cos }^4\theta -4{\cos }^2\theta -4{\cos }^2\theta +4{\cos }^4\theta +1\\ =8{\cos }^4\theta -8{\cos }^2\theta +1\end{array}$

Therefore, $\cos 4\theta$ in terms of

  $\cos \theta$ alone is $8{\cos }^4\theta -8{\cos }^2\theta +1$ .