Brilliant suggestion! Even if it's been suggested before, some of these things are good to bring up again for further discussions..
And I honestly think if you implement taping yourself on a regular basis, it can hugely speed up your improvement process. You're trainer can be there yelling things out your doing wrong over and over, but when you see yourself doing them wrong and make the concious thought to stop doing it,,, I think the subject of this post is perfect.. Not necessarily a secret, but a !!HUGE!!LY useful tool for improving yourself if you are really serious about working on all your techniques...
You can guarantee that just about every olympic sport that requires any skill or technique will be filmed by trainers and studied closely with the athlete to eliminate ANY flaws in their technique....

Also Football games and things like that.. Surely coaches go over video's because their is no other way that you could really show and demonstrate to a team what they are doing wrong, and in a way that they can see and understand immediately with a clear picture. literally...

Props on reminding everyone of this tip!!! I'm going to dust my camera up this afternoon. It's also a bit of fun watching it back later of course...

If you feel silly setting up the camera at the gym, you should offer to film another couple of people for a short time and just play it back for them at the gym. They'll see really quickly what a good idea it is and other will probably wish to bring their cameras in after that.