Then he very clearly is only sorry he got caught, not for doing whatever it was he was lying about.

An example: my wife and I dated briefly in high school.  We broke up, but when we bumped into each other several years later we rekindled and got back together.  At the beginning, she admitted she had been cheating on me during the last few months of our relationship in high school.  She apologized and told me about how shitty of a thing it was to do to me, and she wanted us to start fresh without any secrets.  If she hadn't told me I'd have never known.

Every day for the past ten years she has worked to make up for that mistake.  I have never demanded anything special of her because of it (I was kind of a shithead in high school to be fair), I have never doubted her or not trusted her because as far as I'm concerned it was a different relationship between two different people.  I have unfortunately, used it a handful of times during bad fights, and every time I have I have immediately regretted it because I can see the shame, guilt, and despair in her eyes.  She'll get quiet and look down at the ground.

The point is you can tell when someone is genuinely regretful of their actions versus regretful of just being caught.  If your boyfriend/husband/whoever is making only token efforts to please you, then he's the latter and not the former.  Someone who feels the former is unlikely to reoffend, the latter will probably go right back to doing whatever it was as soon as they're in the clear.