Using the formalism of supersymmetric quantum mechanics, we give an exact solution for a family of onedimensional periodic potentials, which are the supersymmetric partners of the potential proportional to the trigonometric function cos(2x) such that the Schrodinger equation for this potential is named the Mathieu equation mathematically.We show that the new potentials are distinctly different from their original ones. However, both have the same energy band structure. All the potentials obtained in this paper are free of singularities.