ISL 1982 Problem 34

Let $M$ be the set of all functions $f$ with the following properties:(i) $f$ is defined for all real numbers and takes only real values.(ii) For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$ (iii) $f(0) \neq 0.$ Determine all functions $f \in M$ such that**(a)** $f(1)=\frac 52$ ,(b) $f(1)= \sqrt 3$ .