How do I prove that $\cos(\frac{1}{2}x)$ is a periodic function?

Question

Given: $f(x)=\cos(\frac{1}{2}x)$.

Prove: f is a periodic function with period 4π

My math teacher never went over this so I don't know where to start or what to do :/

Answer 1

Hint: In order for $f$ to be $4\pi$-periodic, it is necessary and sufficient to prove that

$$f(x + 4\pi) = f(x)$$

for any choice of $x$. Use the fact that

$$\cos(x + 2\pi) = \cos(x)$$ for any $x$.