Question

Double-angle identities: Use double-angle identities to rewrite each of the following trigonometric expressions until no exponents are involved.

${\cos }^{2}x=\_\_\_\_$

Short answer

$\frac{\cos 2\theta +1}{2}={\cos }^2\theta$

Step-by-step solution

Step 1. Given information.

Consider the given information.

${\cos }^{2}x$

Step 2. Explanation.

Write the given trigonometric expression in cosine form.

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