Both complacent and complaisant descend from Latin complacere, “to please, to be pleasant,” but they have acquired different meanings in English.
complacent (adjective): feeling or showing pleasure or satisfaction, especially in one’s own conditions; self-satisfied.
complacence/complacency (noun): the state of being pleased.
The nouns complacence and complacency convey the idea of self-satisfaction accompanied by a lack of awareness of a potential danger. Here are some examples of current usage:
The danger is that being too complacent can derail your career.
10 Signs Your Employees Are Growing Complacent In Their Careers
Complacent investors have a way of pushing markets steadily higher despite the swirl of bad news around them.
The U.S. can’t afford to get complacent about obesity
Complacency toward Ukraine crisis could cost Conservatives at the polls
complaisant (adjective): the action or habit of making oneself agreeable.
complaisance (noun): courtesy, politeness.
The noun complaisance entered the language by way of French about two hundred years after complacence. It retains its French spelling and the meaning of being pleasant to others. A complaisant person is eager to please.
I think of complaisant as an old-fashioned Jane Austen word, but it is still to be found in current usage:
He [Donald Sterling] will find a complaisant television talk show host to give him a platform for a heartfelt public apology.
By closing down one plant and punishing its workers, workers in other plants would be forced into more complaisant behavior.
As inevitably happens with such similar word pairs, complacent and complaisant are often misused. The following examples use complaisant in the sense of complacent:
“The law doesn’t affect me, so why should/would I care?”–That is a very complaisant attitude.
I think Don has been somewhat complaisant. He’s keeping tabs on the agency but he has made no effort to try and get back in to work.
People who vote for the incumbent are probably pretty complaisant about what their Senator actually does and stands for on the world stage.
We are very complaisant in this country because we have enjoyed so many years of having relatively peaceful lives.