Interesting question and interesting answer by Attikos "Why would you want to?"
I just love people who answer these questions without ever really answering the question. "Why you would want to" has nothing to do with the question at hand and certainly isn't an answer but another question.
Whether you would want to or not, sometimes you have to trust someone who has broken that trust. Sometimes you have no choice (this President, your elected officials come to mind) and that is a reason why - but how is another problem.
To trust someone who has broken your trust you need to ascertain things about them and yourself. Truth is if someone is caught at breaking your trust it isn't the first time, just the first time they've been caught so they are only deserving of one more chance if any at all.
#1) They must be confronted by you with the transgression & they must show themselves to admit and repent to what they have done. There can be no equivocation, rationalization or excuses for their behavior which in itself would be a continuation of their insincerity. This is mainly for your benefit, not theirs, so that you know where you stand and that you have made an honest attempt to bring all aspects into the light & leave no room for misunderstanding.
#2) You must decide if you can emotionally and practically risk another broken trust no matter how #1 goes and understand that even if you forgive them they ARE likely to transgress your trust again and if they do you must know you can handle it and what comes next. Otherwise you should draw the line at one broken trust and make it clear to them you will not trust them again.
#3) Once a trust has been broken (remember this is seldom the first time just the first time they have been caught), the transgressor honestly admitted to it and repented but then breaks your trust again I feel you have no choice but to never trust them again. You must confront them with your conclusions and make it clear they have been given one chance and failed to live up to their promises. Whatever consequences this decision may have for you, your relationships, your job, your life you must be prepared to never be put in a position to trust this person again even if it means separation, divorce, your job or your dream.
This is a serious consideration in any relationship and anyone who doesn't take breaking of a trust seriously and doesn't deal with it deserves whatever consequences ensue from trusting someone who can not be trusted.