ILL 1985 Problem 96

Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:

(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$ , and

(b) $\lim_{x\to \infty} f(x) = 0$ .