Intermediate Value Theorem help

Question

Let $f$ be a continuous function on $\mathbb{R}$ which is periodic with period $2\pi$. This means $f(t + 2\pi) = f(t)$ for all $t$. Show that there exists $x\in[0,\pi]$ such that $f(x) = f(x + \pi)$.

I know it's an intermediate-value theorem problem. I think I have to take the difference of both sides, but not quite sure. Can anyone help?

Answer 1

Hint: Consider $g(x)=f(x+\pi)-f(x)$ ,$g(0)=-g(\pi)$. If $g(0)=0$ we're done (why ?), otherwise $g$ has a root (why ?) and we're also done (why ?).