This paper introduces a dual-function radar-communication (DFRC) system with cognitive radio capability to tackle the spectral scarcity problem in wireless communications. Particularly, a cognitive DFRC system operates on a spectrum owned by a primary system to simultaneously perform data communication and target tracking with the condition that its interference to the primary users (PUs) is below a certain threshold. To achieve this, an optimization problem is formulated to jointly design the beamforming vectors for both the radar and communication functions in such a way that the mean square error (MSE) of the beam pattern between the designed and desired waveforms is minimized. The optimization problem has the following three constraints: i) the signal-to-interference-plus-noise ratio (SINR) at each data communication user is above a predetermined level; ii) the per-antenna transmit power is maintained at a given level; iii) the interference imposed on each PU is below a certain threshold. Both the semidefinite relaxation and nature-inspired firefly algorithms are proposed in order to search for the optimal solutions to the optimization problem. The simulation results indicate that our proposed algorithms can enable the DFRC system to protect the PUs while simultaneously performing its communication and radar functions.