Here is the full list of all 14 LA Cops trophies - there are 2 bronze, 1 silver, 10 gold and 1 platinum.
Kill 1000 criminals.
Kill 600 criminals.
Kill 400 criminals.
Fully upgrade a Cop and purchase all weapons.
Kill 200 criminals.
Kill 100 criminals.
Arrest all criminals in any level.
Finish a level without any of your Cops getting hit.
Finish any level without losing any Cops.
Only the best of the best earn the right to wear a platinum cop badge.
Complete all the game missions in normal mode.
Complete all 8 main story missions.
Complete all the game missions in Hardcore Mode.
Complete all the game missions in Nightmare Mode.