I've been trying to solve this for a few minutes and checking the rules in my book back and forth but I can't seem to understand why. P(A ∩ B)-P(A ∩ B^c) = P(A)?

here are my calculations in which I need to find P(A)

below are the correction from the book which also says P(A ∩ B)-P(A ∩ B^c) = P(A). I thought it was only P(A ∩ B) + P(A ∩ B^c) = P(A)?