Cosine-sine functional equation on semigroups

Abstract: Let $S$ be a semigroup. We determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*} in terms of multiplicative functions, solutions of the special case $$\varphi(xy)=\varphi(x)\chi(y)+\chi(x)\varphi(y), x,y\in S$$ of the sine addition law, where $\chi:S\to\mathbb{C}$ is a multiplicative function, and also in terms of solutions of the particular case $$\psi(xy)=\psi(x)\chi(y)+\chi(x)\psi(y)+\varphi(x)\varphi(y), x,y\in S$$ of the cosine-sine functional equation where $\chi:S\to\mathbb{C}$ is a multiplicative function and $\varphi:S\to\mathbb{C}$ such that the pair $(\varphi,\chi)$ satisfies the sine addition law.