The cosine and sine addition and subtraction formulas on semigroups

Abstract

The cosine addition formula on a semigroup S is the functional equation $g(xy) = g(x)g(y) - f(x)f(y)$ for all $x,y \in S$. We find its general solution for $g,f \colon S \to \mathbb{C}$, using the recently found general solution of the sine addition formula $f(xy) = f(x)g(y) + g(x)f(y)$ on semigroups. A simpler proof of this latter result is also included, with some details added to the solution.

We also solve the cosine subtraction formula $g(x\sigma(y)) = g(x)g(y) + f(x)f(y)$ on monoids, where $\sigma$ is an automorphic involution. The solutions of these functional equations are described mostly in terms of additive and multiplicative functions, but for some semigroups there exist points where f and/or g can take arbitrary values.

The continuous solutions on topological semigroups are also found.