Complacent means self-satisfied, smug, or contented to a fault. Complaisant, a relatively recent loanword from French, means cheerfully obliging; tending to go along with others
Examples
These writers use complacent correctly:
But they were even more horrified to find that the inhabitants of these societies were not only complacent in giving up their freedoms, but even thrilled by the novelty of a mass produced, mass culture. [The Sensible and the Intelligible]
Henry Miller set out to shock the complacent American bourgeoisie with his heady mix of sex and philosophy. [Guardian]
And these writers use complaisant well:
Is it appropriate to offer a class full of literature that tends to reinforce a complaisant status quo view of a society that is homicidal in many ways without subjecting it to normative and conceptual critique? [A Practical Policy]
But my goat, Cookie, was complaisant and calm. [Chron]
