Cauchy's equation and the cosine equation on an abelian group $G$ are both particular cases of the general equation, in which a compact group $K$ acts on $G$, viz. the cases $K = \{ e\}$ and $K = \mathbb{Z}_2$, respectively. We extend a result due to Chojnacki on operator-valued solutions of the cosine equation to this general equation: We prove that if $f$ takes its values in the normal operators on a Hilbert space $\mathcal{H}$, then $f(x) = \Bigint\nolimits_K U(k\cdot x)\,dk$, $x \in G$, where $U$ is a unitary representation of $G$ on $H$, and $dk$ denotes the normalized Haar measure on $K$. We show that normality may not be needed if $K$ is finite, thereby generalizing a result by Kurepa on the cosine equation.