Here is the full list of all 52 Earth Defense Force 4.1: The Shadow of New Despair trophies - there are 36 bronze, 14 silver, 1 gold and 1 platinum.
All Trophies
Obtained all available weapons.
Destroyed 500 carriers.
Destroyed 20 Argos.
Cleared all stages on Inferno difficulty as an Air Raider.
Cleared all stages on Inferno difficulty as a Ranger.
Destroyed 150 red drones.
Cleared all stages on Inferno difficulty as a Fencer.
Cleared all stages on Inferno difficulty as a Wing Diver.
Defeated 20 Greater Wild Dragons.
Cleared all stages on Hardest difficulty as an Air Raider.
Cleared all stages on Hardest difficulty as a Ranger.
Destroyed 6 motherships.
Destroyed 5000 flying drones.
Cleared all stages on Hardest difficulty as a Fencer.
Cleared all stages on Hardest difficulty as a Wing Diver.
Rescued 50 other players in co-op play.
Destroyed 6 flying-type bug nests.
Defeated 360 Retiarii.
Cleared all stages on Hard difficulty as an Air Raider.
Cleared all stages on Hard difficulty as a Ranger.
Destroyed 6 Brains.
Destroyed 200 large transport ships.
Defeated 40 king spiders.
Defeated 4000 flying-type bugs.
Destroyed 900 Hectors.
Cleared all stages on Hard difficulty as a Fencer.
Cleared all stages on Hard difficulty as a Wing Diver.
Defeated 24 queens.
Destroyed 30 Erginuses.
Defeated 200 Deroys.
Destroyed 2000 flying vehicles.
Cleared all stages on Normal difficulty as an Air Raider.
Air Raider's armor has reached 1000.
Cleared all stages on Normal difficulty as a Ranger.
Defeated 120 shield bearers.
Defeated 24 death queens.
Defeated 2400 Dragons.
Defeated 20,000 ant-type bugs.
Cleared all stages on Normal difficulty as a Fencer.
Fencer's armor has reached 1250.
Cleared all stages on Normal difficulty as a Wing Diver.
Obtained 50% of all available weapons.
Defeated 12,000 spider-type bugs.
Destroyed 200 underground tunnel exits.
Destroyed 12 quadruped fortresses.
Ranger's armor has reached 1000.
Healed another player in co-op play.
Rescued 5 other players in co-op play.
Obtained 10% of all available weapons.
Wing Diver's armor has reached 550.
Reached a height of 200 meters.