Hmmm ... perhaps you could trust but verify him on a couple of things that aren't of much consequence.
This could allow him to begin building up a track record of honesty with you. If he lies about the small stuff, then he certainly can't be trusted on the big stuff.
With that said, when you say he is recovering, it really comes down to whether he is actively working on his recovery (and I mean really working hard on it) or just not drinking.
Alcohol was our solution to our deeper problems. If he isn't working on fixing the deeper problems, the cycle usually repeats.