Cosine subtraction laws.

We study two variants of the cosine subtraction law on a semigroup S. The main objective is to solve g (x y ∗) = g (x) g (y) + f (x) f (y) for unknown functions g , f : S → C , where x ↦ x ∗ is an anti-homomorphic involution. Until now this equation has not been solved on non-commutative semigroups, nor even on non-Abelian groups with x ∗ : = x - 1 . We solve this equation on semigroups under the assumption that g is central, and on groups generated by their squares under the assumption that x ∗ : = x - 1 . In addition we give a new proof for the solution of the variant g (x σ (y)) = g (x) g (y) + f (x) f (y) , where σ : S → S is a homomorphic involution. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]