Thank’s to Powerpyx :
This can only be done in a new game before collecting your first crystal. Each level requires you to find a crystal for Cortex and once you do he no longer gets angry.
Step 1: From the title screen, start a new game. You can keep all existing save games, simply select the option "Disable Auto Save" and nothing will be lost.
Step 2: In the first Warp Room start & quit any level 3 times in a row. Once the level is loaded, press the Options-Button to pause and select 'Quit'. It will immediately end the level without having to make any progress. Each time you do this Cortex will remind you of the crystals that you are must find for him. He will get angrier each time. After quitting 3 levels (can be the same level each time) the trophy will unlock!
Important: Never complete a level / never pick up a purple crystal! After collecting the first crystals the trophy no longer works and you'd have to start a new game again.
Technically, this trophy is missable. But since it can be done in a new game within 5 minutes at any time it really doesn't matter when you do it. I want to point out again that no progress is lost when starting a new game, so you can do this whenever you want. You don't have to make any progress in the game to do this. Thereby it's not really 'missable'.