We present a proof of a transformation identity of a complete elliptic integral of the first kind, presented in a recent publication [O. Ciftja et al., Eur. J. Phys. 30, 623 (2009)], where it was derived from an analytical solution of the electrostatic potential due to a uniformly charged ring. In addition, by calculating the electrostatic potential due to a charged ring with a sinusoidal charge distribution, we obtain a new mathematical identity for the complete elliptic integral of the second kind. We show that these two identities can be derived from the existing mathematical identities of elliptic integrals, proving them in a direct manner.