check the wording of your question.
A cosine curve of the type you stated does not have any asymptotes.
the answer you gave of -pi/8<x<3pi/8
represents one period of the curve.
change y=3cos(4x+pi/2)-3
to y=3cos4(x+pi/8)-3
notice that the curve y = 3cos4x has been shifted pi/8 to the left or -pi/8,
the period is 2pi/4 or pi/2
if we add pi/2 to -pi/8 we get 3pi/8