My wife said she wants me to believe in her. Will this even be possible?

Why are you hanging around?

Seriously.

You have proof of … something. Certainly you have proof of lying, and potentially proof of a coverup. But you already don't trust her. And she's already walking on eggshells around you.

What do you need? Do you need to find hotel room receipts? His hair in your bed? An empty box of condoms?

Playing junior detective isn't doing anyone any good here. If you want to stay married, then find a way to forgive her or at least stop hounding her. And recognize that this will probably continue unless you two get into couples counseling and you both go into it with 100% conviction. Go all in or go home.

If you don't want to stay married, then start making moves in that area. But also recognize that maybe you're overreacting—although I honestly doubt that you are. And recognize that ending a marriage is an expensive proposition. But it's not impossible.

This in-between stuff is for the birds. Staying together in this sort of limbo is unhealthy. And if you claim you're staying together for children, then I got news for you. Apart from the tiniest of infants or children who are extremely mentally challenged, your kids know something is going on. They may not know the details, and they may even blame themselves. But they understand things aren't smooth or perfect. Don't kid yourself and pretend you can conceal it from them and somehow protect them. You can't.

And the jig is up, anyway. They know something's off.

If couples counseling isn't in the cards, then seek it out for yourself. Why? To figure out how to handle your anger and hurt, and make a plan for what'll happen next. Whether you step into the next decades of your lives together or apart, it will help to have a plan.

IANAD

I believe there are apps that can help you to track whether your partner is cheating or not. I can't name them off the top of my head but I know a few of them exist so it might be worth it to do some research on Google Play.

That aside, this looks like a tough call because you can't prove she is cheating without concrete evidence but you can't dismiss your suspicions and blindly trust her until she gives you real reasons to do so either. It is up to you if you want to look more into who this man is and what the nature of her associations with him are but her defensive reaction and secretiveness about it feels like red flags to me. But I suggest you consider your steps carefully before you resort to anything as extreme as a divorce because while you don't need the extra stress you don't have to put up with her immature games either.