What you should do, is since it seems to me, that you have lost someones trust, you should try to give them a little space, and allow them to get over what it is that made them stop trusting you. During that time, it would be best if they even wanted to talk to you, to talk to them, and to show others that they can trust you. In one way or another, word will get to them, that you are trustworthy.If you are are still friends with them, you should try your best to help them, and support them.

If you aren't still friends with them, you should just try being a friend, and asking them if they need help, and if they do helping them. Or if they look like they are sitting by themselves, then go and talk to them, and just be a friend to them.

Last, if they were a true friend, then they would not of completely lost trust in you, so if you were really good friends, the general knowledge of how good you were would come back to them, as you are trying to be nice. 

What not to do:

What you should not do, is force them to trust you. This has several effects, they don't trust you to mean what you say, and it pushes them farther away, along with many other things.

Another thing that you shouldn't do is to talk about them behind their back, or blackmailing them. I can't figure out why anyone would do this, but it really doesn't help you at all. Right along with blackmail, you should never just do something out of revenge, or on the spur of the moment, because again, it would make them stop trusting you.