In this paper, we investigate the physical-layer security of a secure communication in single-input multiple-output (SIMO) cognitive radio networks (CRNs) in the presence of two eavesdroppers. In particular, both primary user (PU) and secondary user (SU) share the same spectrum, but they face with different eavesdroppers who are equipped with multiple antennas. In order to protect the PU communication from the interference of the SU and the risks of eavesdropping, the SU must have a reasonable adaptive transmission power which is set on the basis of channel state information, interference and security constraints of the PU. Accordingly, an upper bound and lower bound for the SU transmission power are derived. Furthermore, a power allocation policy, which is calculated on the convex combination of the upper and lower bound of the SU transmission power, is proposed. On this basis, we investigate the impact of the PU transmission power and channel mean gains on the security and system performance of the SU. Closed-form expressions for the outage probability, probability of non-zero secrecy capacity, and secrecy outage probability are obtained. Interestingly, our results show that the strong channel mean gain of the PU transmitter to the PU's eavesdropper in the primary network can enhance the SU performance.