A d'Alembert type functional equation on semigroups

We treat two related trigonometric functional equations on semigroups. First we solve the $\mu$-sine subtraction law $$\mu (y)k(x\sigma (y))=k(x)l(y)-k(y)l(x),x,y\in S,$$ for $k,l:S\to \mathbb{C}$, where $S$ is a semigroup and $\sigma$ an involutive automorphism, $\mu :S\to \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$, then we determine the complex-valued solutions of the following functional equation $$f(xy)-\mu (y)f(\sigma (y)x)=g(x)h(y),x,y\in S,$$ on a larger class of semigroups.