When I say forgive that means not search his things
No, that would be trusting again. Forgiving means, I'm willing to give you one more chance, under these conditions: he must be an open book. He must turn over to you all account passwords, etc. Phones, car mileage, whatever it takes to make you feel you know where he is and what he's doing. He has NO RIGHT to privacy until YOU feel like giving it to him. If this is an issue, then he is still harboring whatever feelings led him to cheat in the first place, and of course, he needs counseling to address that, anyway.
If you think about a healthy marriage, there are no secrets. None. No need for privacy (except in the bathroom, maybe). Your end of conversations with friends, for example, should be open to discuss with your spouse-only a friend's problems and your responses that would reveal the nature of the friend's problem are "private." Any feelings you transmit to a friend about your own life should be shared with your spouse, probably first. Confiding in a friend when you have marital concerns is actually avoiding the problem, because the only way to deal with them is to talk to your spouse. Yes, you can confide in others too (for support, ideas), but truthfully, you and your spouse should be working on the issues.
Anyone who needs tons of "privacy" probably shouldn't be married. Our thoughts are our own, but holding back thoughts that have to do with the marriage-once we are aware we are doing it and the issues are potentially damaging--is counterproductive.
My marriage was like that--an open book--until I gave up on it. Probably the only mistake I made was in not telling my husband when I realized I was emotionally "done." Maybe that would have given him a chance to repair things, but since by that point I didn't think so, and didn't WANT to repair things, I guess I kept my mouth shut and tried to stay married anyway. Didn't work, by the way.
Cheaters will go on and on about needing privacy, but there is a qualitative difference between "me time" and emotional privacy. If you cannot share ALL of your feelings with your spouse, again, why be married? This does not mean spitting out every passing thought of anger, however; it means good communication about issues that bubble up as recurrent. If a spouse does something a couple of times and it ticks you off, you let them know in a respectful way, "Hey, I need the toilet seat down so I don't fall into it at night." If that does not get a positive response, "I feel really humiliated when I fall into the toilet." When these kind of messages get no response, or mediocre at best, it's time for help, b/c they are only going to get worse. Wish I had known that a long time ago.