This is the image of a part of the problem that I am doing. I understand partial fractions fairly well. To solve for A, you will zero out the other term(s). So, For A, if I let B = 0, I am left with teh following:
$ae^{ix}-ae^{-ix} = A - Aae^{-ix}$
What I do not know is how A = 1. I need the steps in between.