Interdependence of additivity and sine additivity

Let S be a semigroup and K a field. A function \(f:S \rightarrow K\) $f:S\to K$ is additive if \(f(xy) = f(x) + f(y)\) $f(xy)=f(x)+f(y)$ for all \(x,y \in S\) $x,y\in S$ , and functions \(g,h:S \rightarrow K\) $g,h:S\to K$ form a sine pair if they satisfy the sine addition law \(g(xy) = g(x)h(y) + h(x)g(y)\) $g(xy)=g(x)h(y)+h(x)g(y)$ for all \(x,y \in S\) $x,y\in S$ . Adding these two equations we arrive at the functional equation (*) \(f(xy) + g(xy) = f(x) + f(y) + g(x)h(y) + h(x)g(y)\) $f(xy)+g(xy)=f(x)+f(y)+g(x)h(y)+h(x)g(y)$ . The alienation question for additivity and sine additivity asks whether (*) implies that f is additive and (g, h) is a sine pair. To fully answer this question we find the general solution of (*) for unknown functions \(f,g,h:S \rightarrow {\mathbb {C}}\) $f,g,h:S\to \mathbb{C}$ . The solution illustrates a significant amount of interdependence between additivity and sine additivity.