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The cognitive radio (CR) network is a promising network architecture that meets the requirement of enhancing scarce radio spectrum utilization. Meanwhile, reconfigurable intelligent surfaces (RIS) is a promising solution to enhance the energy and spectrum efficiency of wireless networks by properly altering the signal propagation via tuning a large number of passive reflecting units. In this paper, we investigate the downlink transmit power minimization problem for the RIS-enhanced single-cell cognitive radio (CR) network coexisting with a single-cell primary radio (PR) network by jointly optimizing the transmit beamformers at the secondary user (SU) transmitter and the phase shift matrix at the RIS. The investigated problem is a highly intractable due to the coupled optimization variables and unit modulus constraint, for which an alternative minimization framework is presented. Furthermore, a novel difference-of-convex (DC) algorithm is developed to solve the resulting non-convex quadratic program by lifting it into a low-rank matrix optimization problem. We then represent non-convex rank-one constraint as a DC function by exploiting the difference between trace norm and spectral norm. The simulation results validate that our proposed algorithm outperforms the existing state-of-the-art methods.