A functional equation motivated by some trigonometric identities

We deal with the following functional equation:f(xy)+λf(x)f(y)=ϕ(x,y), where λ is a complex number and f and ϕ are complex mappings defined on a semigroup. Moreover, we assume an addition formula of trigonometric type for ϕ. Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003.