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 $x$, $y$.
Is it true that if $f(x)$ is not identically zero, and if $|f(x)| ≤ 1$ for all $x$, then $|g(y)| ≤ 1$ for all $y$?