We investigate a special class of nonlinear periodic waves and their stabilities for a Bose–Einstein condensate in a pure cosine nonlinear lattice. It is shown that for a specific range of the lattice parameters, we can obtain two different types of exact periodic waves, one-period and double-period solutions. Landau and dynamical instabilities of these extended states are analyzed. The stability phase diagrams for these exact solutions are identified numerically. Our results indicate that as the modulation amplitude of the nonlinear lattice is increased, the double-period solutions are more stable than the one-period solutions. •For certain specific lattice parameters, we obtain new exact one-period and double-period solutions for a Bose–Einstein condensate in a pure cosine nonlinear lattice.•The stability phase diagrams for Landau and dynamical instabilities of these exact periodic waves are identified numercially.•For a large modulation amplitude of the nonlinear lattice, the double-period solutions are more dynamically stable than the one-period solutions.