Let S be a semigroup. We describe the solutions f, g: S→ C of the functional equation f(xy)=f(x)g(y)+g(x)f(y)-g(x)g(y),x,y∈S,in terms of multiplicative functions on S and solutions of the particular case φ(xy)=φ(x)χ(y)+φ(y)χ(x),x,y∈S,of the sine addition law in which χ: S→ C is a non-zero multiplicative function and φ: S→ C.
