I saw a similar thread for the PS3 version, but not B:AC on the 360...
I have duplicated this, once playing from the start without installing the game on my 360's hard drive, and once with the game installed.
Inside Zsasz' hideout, there are two Riddler Trophies, and one Riddle. The Riddle concerns The Broker, and is answered by scanning a pile of cards on Zsasz' desk, once you defeat him.
One trophy is attached to a wall, in a bend just out of sight to the right of the first gap between platforms, once you get inside the building (this gap is the only one that has a bridge, which floats on the surface of the water). By freezing an ice platform with the Ice Grenade, it is easy to reach.
The other trophy is in a cage. Once you reach Zsasz and defeat him, walk up to his desk, and activate the console (the game will display the "Use" prompt with the 'A' button when you are standing in front of it). An electrical spark will begin under the bridge leading from the front door to Zsasz platform. The cage releases when you electrify a Remote Batarang by flying it through the spark, and into a vent which contains a fuse box. The fuse box will short (you will see smoke coming from it), and the cage opens.
IF YOU FALL INTO THE WATER before leaving the building, while moving towards that trophy, you will trigger the glitch. If you fall into the water, even if you recover, grab the trophy, and then leave, the trophy will **never** count as being collected. You will have to restart the game in order to collect all the Riddler trophies. I have tried copying my saved game over to a memory card, but it doesn't help. I don't know how to resolve this problem, and if you waited to pursue the Riddler until after the main story mission, then you have a lot to cover. At least I can document exactly what causes this.
How do you avoid falling into the water to get back to that trophy? There's that point where you have to toss the Freeze grenade under a pipe, then run and slide under the pipe onto the ice raft, but how would you get through that section the other way? The only way I could do it was to fall in the water, then Batman climbed up on the other side of the room. I think it still counted for me, but I'm not worried about that. I'm just curious how you would get back under that pipe without falling in the water.
You abandoned your friend to die. He's calling for you, while the children spread out his heart to the skies.
So that's how you're supposed to get the cage trophy? I ended up using the Line Launcher in the starting area and having Batman perch on top of the line. From there, I shot the Remote Electrical Charge through the duct at the fuse box.
Getting the line launcher line at the exact correct height was a bit of a pain...