Are these conditional probability statement true?

Question

Let $A$ and $B$ be events with $A>0$ and $B>0$. Are the following 2 statements true:

1. $P(B|A^c)=(P(B)P(A^c|B))/(P(A^c))$

2. $P(A|B^c)=1−P(A^c|B^c)$

I believe they're both correct but the way in which they interchange the complements makes me doubt whether it really is correct or not.

Answer 1

$$P(B|A^c)=\frac{P(B\cap A^c)}{P(A^c)}=\frac{P(A^c|B)P(B)}{P(A^c)}$$

Now $B^c\cap A$ and $B^c\cap A^c$ are disjoint sets and $(B^c\cap A)\cup(B^c\cap A^c)=B^c$, hence $P(B^c)=P(B^c\cap A)+P(B^c\cap A^c)$ $$P(A|B^c)=\frac{P(A\cap B^c)}{P(B^c)}=\frac{P(B^c)-P(B^c\cap A^c)}{P(B^c)}=1-\frac{P(B^c\cap A^c)}{P(B^c)}=1-P(A^c|B^c)$$

But for the above probabilities to be defined we should have $P(A^c),P(B^c)>0$.