This letter studies the physical layer security of an unmanned aerial vehicle (UAV)-enabled multicasting system, where a UAV serves as a mobile transmitter to send a common confidential message to a group of legitimate users under the existence of multiple eavesdroppers. The worst situation in which each eavesdropper can wiretap all legitimate users is considered. We seek to maximize the average secrecy rate by jointly optimizing the UAV's transmit power and trajectory over a given flight period. The resulting optimization problem is nonconvex and intractable to solve. To circumvent the nonconvexity, we propose an iterative algorithm to approximate the solution based on the alternating optimization and successive convex approximation methods. Simulation results validate the convergence and effectiveness of our proposed algorithm.