Five Trigonometric addition laws on semigroups

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