Trust is earned.
If your SO has betrayed your trust in any way, then the burden becomes theirs to earn your trust again. If they want to stay and earn your trust again, they have to be willing to put up with a certain amount of ****.
But the burden is also yours to decide one of two things:
1) Is the difficult road that lies ahead (rebuilding trust) acceptable to you? Is the relationship worth salvaging? Is this the first time or the 100th time your trust has been betrayed? If you accept the task of rebuilding and decide to stay you are also accepting the fact that it will be very uncomfortable at times.
2) if you stay then it means you are willing to TRY to learn to trust them again. There are times you will want to pin them to the wall with questions and accusations but you'll have to stuff those feelings instead. After all, you have decided to stay in the relationship and give it a fair chance. This means 1/2 of the responsibility is yours to be and act in a reasonable manner.
That being said, however, you can always change your mind if you've concluded that it's just not possible for you. It is then time to detach and move on.
None of it is easy. That's why it is said that trust is the most important issue in a relationship...because once that is gone it is almost impossible to get it back.