The cosine addition and subtraction formulas on non-abelian groups

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We determine the solutions f,g,h∈C(G) of the Levi-Civita equation (Formula presented.) that extends the cosine addition law. As a corollary we obtain the solutions f,g∈C(G) of the cosine subtraction law g(xy∗)=g(x)g(y)+f(x)f(y), x,y∈G where x↦x∗ is a continuous involution of G. That x↦x∗ is an involution, means that (xy)∗=y∗x∗ and x∗∗=x for all x,y∈G.