Cosine and Sine addition and subtraction law with an automorphism

TL;DR

This paper characterizes complex solutions to certain functional equations involving automorphisms on semigroups, revealing their equivalences and providing applications in the context of cosine and sine addition and subtraction laws.

Contribution

It describes solutions to new functional equations involving automorphisms on semigroups and establishes their equivalences, extending classical trigonometric addition laws.

Findings

Solutions are characterized for the given functional equations.

The first two equations are shown to be equivalent to their variants.

Applications of the results are provided.

Abstract

Let $S$ be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations \[g(x\sigma (y)) = g(x)g(y)+f(x)f(y),\ x,y\in S,\] \[f(x\sigma (y)) = f(x)g(y)+f(y)g(x),\ x,y\in S,\] and \[f(x\sigma (y)) = f(x)g(y)-f(y)g(x),\ x,y\in S,\] where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.