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arXiv · 2302.10263

Cosine and Sine addition and subtraction law with an automorphism

Youssef Aserrar·Elhoucien Elqorachi
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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.

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Youssef Aserrar, Elhoucien Elqorachi. 2023-02-20. Cosine and Sine addition and subtraction law with an automorphism. https://arxiv.org/abs/2302.10263

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