  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.
