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 N. Cortex.
Earn 99 lives.
Defeat N. Gin.
Die as Crash.
Defeat Tiny Tiger.
Discover a Death Route.
Defeat the Komodo Brothers.
Fly, then die.
Unlocked the Secret Exit in Air Crash.
Defeat Ripper Roo.
Discover a "?" Bonus Path.
Collect all the Crash Bandicoot 2: Cortex Strikes Back trophies.
Discover a Gem Path after earning a Color Gem.
Unlocked the Secret Exit in Bear Down.
Unlocked the Secret Exit in Diggin' It.
Unlocked the Secret Exit in Unbearable.
Unlocked the Secret Exit in Hangin' Out.
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.
Just "forget" it. Repeatedly.
Defeat 5 enemies with Coco's Double Leg Drop.