It is known that a pair (f, g) of functions with f≠ 0 satisfies the sine addition formula f(xy) = f(x) g(y) + g(x) f(y) on a semigroup only if g= (μ1+ μ2) / 2 where μ1 and μ2 are multiplicative functions. Here we solve the variant f(xy) = g1(x) h1(y) + g(x) h2(y) for four unknown functions f, g1, h1, h2 on a monoid, where g is not simply the average of two multiplicative functions but more generally a linear combination of n≥ 2 distinct multiplicative functions.
