Given that $f(x)$ is differentiable, $$f(2x)=2(f(x))^2 -1$$
I can assume that $f(x)=cos(ax)$, but I can’t think of a way to prove that it is the only solution. In fact, I am not sure that $f(x)=cos(ax)$ is the only solution. All i know is that $f(0) = -{1 \over 2}$ or $1$.