Tour Eiffel can be photographed from either Trocadero or from Champ de Mars. I am not sure which is the popular destination for tourists. As for me, I always take my friends to Trocadero from where the view is breathtaking. If you choose the (Metro) exit carefully, your unsuspecting friend will never know there is a great view until you ask him to turn right or left. Your friend will be speechless as soon as he/she sees Tour Eiffel and that silence is your reward.
I have never photographed Tour Eiffel from Champ de Mars. On many previous occasions, I have witnessed countless tourists doing the weird hand techniques (like what you see in front of Leaning Tower of Pisa) over here. (Now, stop smirking at the usage tourist and move on. I'm a local now!). On Saturday, I walked over to this side with a photographer friend. There is a structure nearer to the Ecole Militaire side which gave me a opportunity to shoot "picture-in-picture" style. I wanted a human being inside the frame. My friend obliged.