To derive an identity for [tex]\(\cos^2 \theta\)[/tex] using the given cosine double-angle identity, follow these steps:
1. Start with the cosine double-angle identity:
[tex]\[
\cos(2\theta) = 2 \cos^2 \theta - 1
\][/tex]
2. Rearrange the identity to solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos(2\theta) + 1 = 2 \cos^2 \theta
\][/tex]
3. Isolate [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = \frac{\cos(2\theta) + 1}{2}
\][/tex]
Thus, the identity for [tex]\(\cos^2 \theta\)[/tex] derived from the cosine double-angle identity [tex]\(\cos(2\theta) = 2 \cos^2 \theta - 1\)[/tex] is:
[tex]\[
\cos^2 \theta = \frac{\cos(2\theta) + 1}{2}
\][/tex]
