Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(ax+by) +f(ay-bx) = (f(a) +f(b)) (f(x) +f(y)) $
for all values of a, x, b, y.
I was able to show that $f(x) = 1/2$ and $f(x) =x^2$ are solutions but I can't show if they are the only ones. I had the feeling that Brahmagupta-Fibonacci identity would be used here looking at it but I wasn't able to use it.