Actually, the proper way to kill him is to wait for the Sarge to shoot him three times in a row with his beam rifle, this will lower his shields allowing you a second or so to hit him with whatever weapons you have ready.
Otherwise, you can kill him by sheer firepower, but he takes little or no damage while his shield is up, and it takes a ton of damage at one time to knock down his shield without the sarge's help.
It's possible to get Tartarus to fall off the side of the platform, however he doesn't stay dead. Instead the game respawns him at the bottom of the lift in the middle.
I've beaten him on legendary difficulty, and let's just say he was easier than the Brutes that would spawn after you damaged him enough.
My recommendation for anyone doing this on legendary is to lead him around making sure the sarge has a clear shot to Tartarus, then throw a plasma grenade on him as soon as the sarge starts shooting him. The grenade will detonate while his shields are down, doing a lot of damage on top of you shooting him. If he gets too close, go invisible and jump away. You can also camp on the upper level and shoot Tartarus from there. Staying on the upper level also has the advantage of avoiding the brutes that spawn.
After you get a good solid hit the first time his shields drop, go invisible and make a beeline to the lift to get to the top level. Brutes armed with ranged weapons spawn, and those are more of a threat in Legendary. Just stay on the top level, using your carbine and invisibility to get perfect headshots on them, don't hesitate to retreat if they even look your way, especially if they have that grenade launcher. One hit from that will knock out your shields and the other will kill you; they can, and will, rapidly fire two at you.
There's two, maybe three waves of brutes that are spawned when you do enough damage to Tartarus, so always check to see if brutes are incoming from the other passage (the one the sarge isn't on) after you hurt Tartarus.
After that, all you really need to do is make sure Tartarus doesn't get close to you, and don't get caught on the level geometry.