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Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g(xσ(y))=g(x)g(y)+f(x)f(y),x,y∈S,f(xσ(y))=f(x)g(y)+f(y)g(x),x,y∈S,\matrix{ {g\left( {x\sigma \left( y \right)} \right) = g\left( x \right)g\left( y \right) + f\left( x \right)f\left( y \right),} & {x,y \in S,} \cr {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) + f\left( y \right)g\left( x \right),} & {x,y \in S,} \cr } and f(xσ(y))=f(x)g(y)-f(y)g(x),x,y∈S,\matrix{ {f\left( {x\sigma \left( y \right)} \right) = f\left( x \right)g\left( y \right) - f\left( y \right)g\left( x \right),} & {x,y \in S,} \cr } where σ : S → 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.

Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and σ: S → S an automorphism that need not be involutive. We determine the complex-valued solutions of the following functional equation f(xy) - µ(y) f(σ(y)x) = 2f(x)g(y), x,y ∈ S, where µ: S→C is a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. This enables us to solve the functional equation f (xφ(y))-f(ψ(y)x) = 2f(x)g(y), x,y ∈ S, where φ, ψ: S → S are automorphisms such that φ is involutive and ψ is not necessarily involutive. Some consequences of these results are presented. [ABSTRACT FROM AUTHOR]