Sine and cosine addition equations

Summary

The sine and cosine addition equations are the system of functional equations $S(x + y) = S(x) C(y) + C(x) S(y)$ and $C(x + y) = C(x) C(y) - S(x) S(y)$ for two unknown functions $S, C \colon \R \to \R$. They turn the angle addition formulas of sine and cosine into equations. They are equivalent to Cauchy's exponential equation for the complex-valued function $C + iS$. So they characterize sine and cosine, up to a scaling of the variable and an exponential factor, among continuous functions.