Abstract
Let S be a semigroup and K be a field. In a recent article we introduced a new cosine functional equation g(xyz) − g(x)g(yz) − g(y)g(xz) − g(z)g(xy) + 2g(x)g(y)g(z) = 0 for an unknown function g : S → K. It was shown that this equation is closely connected to the sine addition formula, and for K = ℂ its solutions are expressible in terms of multiplicative functions. Here we solve the more general functional equation f(xyz)+g(x)g(yz)+g(y)g(xz)+ g(z)g(xy) + h(x)h(y)h(z) = 0 for three unknown functions f, g, h : S → ℂ, where S is a monoid. The solutions are linear combinations of two multiplicative functions.