To earn the trophy that's called "Mano-A-Mano", you will need to defeat a Titan Henchman without using Batarangs. This can be done on any play mode, in either story or challenge modes.
I would recommend the easier version of the map "Rumble In The Jungle". You'll need to find enough Riddler trophies or rather a specific one to unlock it.
Make your way to the fourth round of this challenge and you'll encounter a group of thugs and a Titan Henchman. From start until the very end, do not take on the Titan and work on incapacitiaing all the thugs in the room.
When all the thugs are done, take out your Explosive Gel and quickly set it down a little before the Titan henchman. Make sure you are holding down
. Just as he is about to step over it as he is charging you, then hit
and the explosive should go off, making him dizzy for a few moments. This is the perfect time to wail on him with your regular punching. Keep punching at him until Batman performs his finishing move on the Titan henchman.
Once the screen loads up with your score results and with Joker talking, the trophy should be unlocked.
The video solution that comes from WikiGuides shows you not only how to earn the trophy, but it also shows you the Riddler trophy you need to get to unlock Rumble In The Jungle, which can be found in the Main Sewer Junction.
Happy hunting and good luck!