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