During drilling operations, drillers are susceptible to unsafe cognitive states, including fatigue, distraction, anxiety, and cognitive conflict, owing to prolonged high-intensity workloads. These cognitive states are major contributors to human errors and safety accidents. However, existing electroencephalogram (EEG)-based studies have primarily focused on recognizing a single cognitive state, making them inadequate for the accurate identification of multiple cognitive states in complex operational environments. To address this limitation, a cognitive state recognition method integrating EEG rhythmic wave power and brain functional connectivity features was developed. A simulated drilling experiment was conducted to collect EEG signals from 17 drillers under five cognitive states. Rhythmic wave power features were extracted from the midline electrodes (Fz, Cz, and Pz) across five frequency bands. Meanwhile, brain functional connectivity matrices were constructed using 32-channel EEG signals by combining the weighted phase lag index (WPLI) and phase transfer entropy (PTE). Leave-one-subject-out cross-validation (LOSO) was employed for feature evaluation and performance assessment. The results show that the proposed method achieves an average recognition accuracy of over 90% for five cognitive states, significantly outperforming methods based on a single feature. Moreover, the selected Fz, Cz, and Pz electrodes provide a practical basis for the development of lightweight and portable EEG devices, offering technical support for real-time monitoring, risk early warning, and engineering implementation of multi-cognitive state recognition in drilling operations.