We study an inverse spectral problem for the radial Schrödinger operators on the unit interval. This problem consists in the recovery of the potential on a subinterval $(0, a)$, $a\leq 1$, from eigenvalues corresponding to the boundary value problems with different boundary conditions. We obtain a sufficient condition for the unique specification of the radial Schrödinger operator by a set of eigenvalues and a part of the potential function on $(a, 1)$ in terms of the cosine system closedness. The Borg-type and the Hochstadt-Lieberman type results are obtained as corollaries of our main result. The main tool of our proof technique is the singular transformation operator representation for the solution of the radial Schrödinger equation.