A generalization of the cosine addition law on semigroups

Our main result is that we describe the solutions $g,f:S\to \mathbb{C}$ of the functional equation $$g(x\sigma (y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma (y)),x,y\in S,$$ where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed constant and $\sigma :S\to S$ an involutive automorphism.