Here is the full list of all 25 Crash Bandicoot 2: Cortex Strikes Back trophies - there are 7 bronze, 11 silver, 6 gold and 1 platinum.
Defeat 5 enemies with Coco's Double Leg Drop.
Unlocked the Secret Exit in Hangin' Out.
Discover a Gem Path after earning a Color Gem.
Die as Crash.
Defeat Tiny Tiger.
Discover a Death Route.
Unlocked the Secret Exit in Bear Down.
Defeat the Komodo Brothers.
Defeat Ripper Roo.
Discover a "?" Bonus Path.
Collect all the Crash Bandicoot 2: Cortex Strikes Back trophies.
Defeat N. Gin.
Defeat N. Cortex.
Unlocked the Secret Exit in Air Crash.
Unlocked the Secret Exit in Diggin' It.
Unlocked the Secret Exit in Unbearable.
Earn 21 Gems.
Earn 42 Gems.
Earn 5 Relics.
Earn 10 Relics (gold or better).
Earn 27 Relics (gold or better).
Don't feel guilty about jumping on him.
Fly, then die.
Just "forget" it. Repeatedly.
Earn 99 lives.