I'm struggling with this former Putnam Exam problem:
Suppose $f$ and $g$ are nonconstant, differentiable, real-valued functions on $R$. Furthermore, suppose that for each pair of real numbers $x$ and $y$, $f(x + y) = f(x)f(y) - g(x)g(y)$ and $g(x + y) = f(x)g(y) + g(x)f(y)$. If $f'(0) = 0$, prove that $(f(x))^2 + (g(x))^2 = 1$ for all $x$.
Right. So obviously, $f(x) = \cos x$ and $g(x) = \sin x$ satisfy the conditions and also the conclusion of the problem. But are these the unique such functions, and if so, how to prove it? And if not, then how to prove the conclusion otherwise?