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I'm guessing you must be new to DLC Trophies in general, regardless of if you have Full Burst or not DLC trophies will appear in your trophies list, this is the same for any game that has DLC trophies.
Oh and I almost forgot, DLC trophies are not required for Platinum, so say you got a game that's been out for awhile that has DLC trophies, you only need to get the trophies that came with the game (IE: all the trophies without the '+') to achieve platinum in that game.
There is a Ghost in my attic and he says: "It's very dark up here...." ....Oh right....and he also says: "Boo."