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.
Unlocked the Secret Exit in Air Crash.
Defeat N. Gin.
Discover a Gem Path after earning a Color Gem.
Unlocked the Secret Exit in Hangin' Out.
Die as Crash.
Defeat Tiny Tiger.
Discover a Death Route.
Unlocked the Secret Exit in Bear Down.
Defeat the Komodo Brothers.
Fly, then die.
Don't feel guilty about jumping on him.
Defeat Ripper Roo.
Discover a "?" Bonus Path.
Collect all the Crash Bandicoot 2: Cortex Strikes Back trophies.
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).
Just "forget" it. Repeatedly.
Defeat 5 enemies with Coco's Double Leg Drop.
Earn 99 lives.