Well finally right after posting it came to me, I started with (A*A)^-1 and it worked out pretty simply from there.
There is a matrix that is normal and invertible. I need to show that its inverse is also normal.
So I know if a matrix is normal then A*A=AA* and if a matrix is invertible then A^-1A=I. So I will need to show that A^-1(A^-1)*=(A^-1)*A^-1. However I'm not sure if it is because it is late, but I'm having a bit of difficulty coming up with the initial step. Any hints would be appreciated.