How to find Maclaurin series of $cos^2(x)$

Question

How to find Maclaurin series of $\cos^2(x)$ from $\cos(x)$.

Maclaurin Series of $\cos(x) = 1-\frac{x^2}{2}+.....$

Then how to find for $\cos^2(x)$ from $\cos(x)$.

For the $cos^2(x)$, I thought of just squaring the right hand side.

Please explain me the method.

Answer 1

It becomes far easier if you invoke this trig identity. $$\cos^2(x) = {1 + \cos(2x)\over 2}.$$