When you say that you cannot see playing video does that include "non-full screen mode", ie, does any DVD's/video playing within a mediaplayer window not show.
I'm not exactly sure of your setup (please specify more) but I'm guessing that if all is fine on your standard Monitor, it could be something to do with any anti-pirating measures that are often built into media programs and drivers.
Whats your specs and display setups, as well as the software you are using. I know that ATI in the past made you choose which display to use for video - Monitor or external TV, so I'm guessing this problem is along those lines.
Still, give me the extra info and while your at it, check to see if such driver options are availible for you to solve your problems. If it is absolutely nessesary for you to display video on both displays at the same time, I'm guessing one option would be to use modified drivers/software, but that cannot be discussed here.