Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions:
(a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$;
(b) $\lim\limits_{x\to\infty}f(x) = 0$.
I found this problem in IMO 1985 longlist. I have been able to figure out that there is one solution of the form $f(x) = 0$ for all $x$.
Any other function satisfying the above has to have $f(0) = 1$ and $f(x) = f(-x)$.
However I don't know how to proceed.
Any thoughts?