A system of cosine-sine functional equations on a semigroup generated by its squares

Abstract: Given a semigroup $S$ generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations \begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+\lambda_{1}^{2}\,h(x)h(y),\; x,y\in S,\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+\lambda_{2}^{2}\,f(x)f(y),\; x,y\in S, \end{align*} where $\lambda_{1},\lambda_{2}\in\mathbb{C}$ are given constants and $f,g_{1},g_{2},h:S\to\mathbb{C}$ are unknown functions.