I wrote a set of APIs which will consider you authenticated if you logged on the website (and therefore you have your cookies set).
I then wrote a Greasemonkey and a Google Chrome plugin that does different calls to my api/* calls. However, Ajax doesnt send cookie header over cross domain (remember it is plugin that is enabled when you are on facebook).
What are the best strategies to authenticate my user and authorize his api calls?
indeed does an AJAX-call and sends the cookie header.
You can therefore simply call a website that returns the authentification status based on visitors cookies, return a simple result and use it.
Example