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 \(f,g,h \in C(G)\) $f,g,h\in C(G)$ of the Levi-Civita equation \(\begin{aligned} g(xy) = g(x)g(y) + f(x)h(y), \ x,y \in G, \end{aligned}\) $$\begin{array}{c}\hfill g(xy)=g(x)g(y)+f(x)h(y),x,y\in G,\end{array}$$ that extends the cosine addition law. As a corollary we obtain the solutions \(f,g \in C(G)\) $f,g\in C(G)$ of the cosine subtraction law \(g(xy^*) = g(x)g(y) + f(x)f(y)\) $g(x{y}^{\ast })=g(x)g(y)+f(x)f(y)$ , \(x,y \in G\) $x,y\in G$ where \(x \mapsto x^*\) $x\mapsto x^{\ast }$ is a continuous involution of G. That \(x \mapsto x^*\) $x\mapsto x^{\ast }$ is an involution, means that \((xy)^* = y^*x^*\) ${(xy)}^{\ast }=y^{\ast }{x}^{\ast }$ and \(x^{**} = x\) $x^{\ast \ast }=x$ for all \(x,y \in G\) $x,y\in G$ .