The author gives the following result: Let \(X\) be a real Banach space and let \(K\) be a closed convex cone in \(X\) such that \(\text{int\,}K\neq\emptyset\). Assume that \(\{F_t: t\in\mathbb{R}\}\) is a regular cosine family of continuous linear multifunctions \(F_t: K\to cc(X)\) and \(x\in F_t(x)\) for all \(x\in K\) and \(t\in\mathbb{R}\). Then there exists a continuous linear multifunction \(H: K\to cc(K)\) such that \[ F_t(c)\subset \sum^\infty_{n=0} {t^{2n}\over (2n)!} H^n(x) \] for \(x\in K\) and \(t\in\mathbb{R}\).