IMO 1972 Problem 5

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$ . $f$ is not identically zero and $|f(x)|\le1$ for all $x$ . Prove that $|g(x)|\le1$ for all $x$ .