New Oxford American Dictionary:
Complacent and complaisant are two words that are similar in pronunciation and that both come from the Latin verb complacere ‘to please,’ but in English do not mean the same thing. Complacent is the more common word and means ‘smug and self-satisfied’: after four consecutive championships, the team became complacent. Complaisant, on the other hand, means ‘willing to please’: the local people proved complaisant and cordial.