I want to display a video on an external monitor while the computer is locked (so that no one can walk up and stop the video, access my account, etc.).
Can I do this in OS X? If not, are there any third party solutions I can use?
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You could create a screensaver using the video file so that it plays while the screen is locked.
If you're code inclined you may even want to try writing it yourself: http://cocoadevcentral.com/articles/000088.php
Or alternatively, there are utilities that will do most of the work for you: http://iscreensaver.com