Using addition/subtraction identities to solve an angle

This is driving me crazy! I understand how to solve an angle such as cos345˚ (the last one I tried) by splitting it into two special angles that add to 345˚. But I almost always get the wrong answer, even though when I follow along with the hints, I'm doing everything exactly the same (I just usually end up picking different angles). I was under the impression that it should work no matter what angles you use, since there doesn't ever seem to be any rhyme or reason to the angles that are picked, or whether addition or subtraction is used, nor are there any suggestions or hints regarding the best way to determine which angles to use (and I have tried searching multiple sites for this information). But for whatever reason, if I don't happen to pick the same angles as are used in the hints, I get a different answer.

For example, with 345˚, I would split that into 300˚ and 45˚ - 45˚ obviously being part of a 45-45-90 triangle, and 300˚ being part of a 30-60-90 triangle (360 - 300 = 60)

There must be something I'm missing! Can someone please help me?