This is arguably the most difficult miscellaneous trophy in the game. Shooting down 45 Japanese Zeros (the planes), even on recruit difficulty, is not an easy task. Don’t focus too much on the PT boats, and don't worry about saving all the sailors, as it’s not required to complete the mission.

If you play the mission enough times, you may be able to memorize where the planes spawn, as they appear to spawn in the same places each time you play the level. There are at least 57 planes that can be shot down, but that's still not a lot of room for error. This will most likely take a few tries, so just keep going at it until you earn this trophy. For your own sake, turn off the rumble feature on your controller, as every shot you fire will make it rumble.
 

Sum of All Zeros Video Guide

 Credit to Kill3rCommando for the video.