simplifying expressions - A puzzling result of "FullSimplify"

When I input:

FullSimplify[(1 - a^2)/b^2, a^2 + b^2 == 1]   

the result is the ideal answer 1；

However，if I input:

FullSimplify[(1 - b^2)/a^2, a^2 + b^2 == 1]   

the result is (1 - b^2)/a^2， instead of the ideal answer 1. How to get the correct answer 1?