Five Trigonometric addition laws on semigroups

Abstract

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