The sine addition law applied to a cosine equation

Abstract Let S be a semigroup and K a field. Ebanks showed recently how the functional equation $$\begin{aligned} g(xyz) = g(x)g(yz) + g(y)g(xz) + g(z)g(xy) - 2g(x)g(y)g(z), \end{aligned}$$ g ( x y z ) = g ( x ) g ( y z ) + g ( y ) g ( x z ) + g ( z ) g ( x y ) - 2 g ( x ) g ( y ) g ( z ) , where $$g:S \rightarrow K$$ g : S → K denotes the unknown function and $$x,y,z \in S$$ x , y , z ∈ S , is related to the sine addition law on S . We simplify Ebanks’ treatment. Furthermore we discuss for solutions $$f,g:S \rightarrow K$$ f , g : S → K of the sine addition law the uniqueness of the component f given g , and we show that f and g are abelian, when $$f \ne 0$$ f ≠ 0 .