So to my understanding, my teacher just told me that the following is incorrect:
If P(D|A^c)=0.4, then P(D^c|A^c) = 1-0.4 which is 0.6?
I was very sure this is true but she just said it's not as I need to work out the intersection and use a formula. Is what I put not correct??
What's written in the post is correct, though of course the instructor might have expected some argument for it.
One such argument would be to remark that $$(D\cap A^c)\cup (D^c\cap A^c)=A^c\implies P(D\cap A^c)+P(D^c\cap A^c)=P(A^c)$$
where we have used the fact that $D\cap A^c$ and $D^c\cap A^c$ are disjoint.
But then we have $$P(D\,|\,A^c)+P(D^c\,|\,A^c)=\frac {P(D\cap A^c)+P(D^c\cap A^c)}{P(A^c)}=1$$
which is the claim that you relied on.