Once trust is lost, it can be very hard to regain. This is not the definitive guide to regaining trust and I am not an expert. This is just my opinion. I believe that there is no situation where trust cannot be rebuilt, but to be honest you the person who broke the persons trust has absolutely no control over the situation other than to just walk away if you do not care.
First off, let the person be angry. Do not expect them to get over that hurt. When they are ready, they will either forgive or not, but if you push it, it will become very hard for them to even move to that point.
When they forgive, they are basically saying they are letting it go, but that does not mean that they trust you. People completely misunderstand forgiveness. They think that the saying forgive and forget means they are not going to remember what happened. What? Can you completely forget when people have hurt you.
When they are ready to talk to you, they will. At that point I suggest you listen. If there is something to say it really should be in the form of a question. If you repeat I understand, I can almost guarantee that they may get even more mad. When they ask you to say something or ask you for your opinion or ask you want you were thinking, etc. I suggest you answer and I do not know is not an appropriate answer. You know why you did what you did to hurt that person to make them not trust you anymore. So fess up and give them the reason.
If you try to blame them, by saying it was your fault this happened, I can again almost guarantee that it will not turn out well. You and only you did this so take responsibility for your actions. If they did something to hurt you and you did not let them know, again that is your fault.
Am I saying that if you let them alone, wait for them to forgive you, then talk when they are ready that they will trust you again, absolutely not. People are not machines that you can write a little code for and poof everything is fixed.
The biggest success for any relationship is by communicating, so communicate before it ever gets to the point where people do not trust.
There is one who will never break your trust and that is the Lord.
Proverbs 3:5-6 New International Version (NIV)
5 Trust in the Lord with all your heart
and lean not on your own understanding;
6 in all your ways submit to him,
and he will make your paths straight.
I pray that if you have broken someones trust or you are the one who broke the trust of another, that you are able to see the error in your ways and try to correct that mistake. I pray that if you do decide to embark on that task, that the Lord provides you with the wisdom to make the right choices and to say the right things. I pray that trust that was broken can be healed and that all people involved will turn to God for guidance. In Jesus name I pray for this, Amen.