A generalization of the cosine addition law on semigroups.

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]