Cosine and Sine Addition and Subtraction Law with an Automorphism

Abstract

Abstract Let S be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations g ( x σ ( y ) ) = g ( x ) g ( y ) + f ( x ) f ( y ) , x , y ∈ S , f ( x σ ( 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 ( x σ ( 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.