It’s easy to mix up these two words, but there’s a world of difference in their meanings.
Complacent means self-satisfied or smug and derives in part from the Latin verb placere (to please). Example:
He felt complacent about his excellent examination results.
Complaisant, on the other hand, means eager to please or obliging. An example of its use is:
He had thought she might oppose him, but found that she was surprisingly complaisant.
Here are some quotations from newspapers:
… announced Thursday along with other Council committee assignments for the next four years, as focused on prodding a “complacent bureaucracy” to better provide city services and dig “deep into the operational failures of city … (www.nytimes.com)
Coming from San Francisco, I had feared that I would find L.A. too complacent about healthcare issues that were important to me. After 20 years working in the San Francisco Health Department … (www.latimes.com)
In fact, the fast way to $100 a share is the kind of deep cost-cutting that this relatively complaisant company has never been known for. This includes, as an order of magnitude, what knowledgeable insiders report is a call to slice $100 million out of CNN’s annual budget of nearly $750 million. — USA Today