Five Trigonometric addition laws on semigroups

Abstract

In this paper, we determine the complex-valued solutions of the following functional equations \[g(xσ(y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)+f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)-f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}\backslash \{0\}$ is a fixed constant and $\sigma :S\to S$ an involutive automorphism.