International Mathematical Olympiad 1972 Problem 5

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $$f(x + y) + f(x - y) = 2f(x)g(y)$$ for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.