We will give a necessary and sufficient condition for the infinitesimal generator of a strongly continuous cosine operator function $C(t)$, such that $\|C(t)\|\le1$ for all $t\in R$ on a reflexive, strictly convex (complex) Banach space with a G\^ateaux differentiable norm to be a spectral scalar type operator with the spectral family of hermitian bounded linear projectors.