Sine and cosine addition equations

Remarks

Example: $S = \sin$ and $C = \cos$ form a solution, by the angle addition formulas of sine and cosine. Non-example: $S(x) = x$, $C(x) = 1$ is not a solution, since the cosine equation gives $1 = 1 - xy$, which fails for $x = y = 1$. Set $E = C + iS \colon \R \to \C$. The two equations together say exactly that $E(x + y) = E(x) E(y)$. So the continuous solutions are $S = C = 0$ and $S(x) = e^{bx} \sin(ax)$, $C(x) = e^{bx} \cos(ax)$ with $a, b \in \R$. Each of the two equations is also studied alone, with one or both functions unknown.