To trust someone, or to trust in someone means that you don’t have to worry, because the person will be honest, do what he should do, not cheat you, be reliable, etc. (“Trust” is more reliable than “trust in”.)
Sometimes to believe in is used with this meaning also, but more frequently it means that someone believes something exists or should be done.
Some people believe in God and some people don’t.
(= Some people believe that God exists and some people don’t.)
Some parents don’t believe in corporal punishment.
(= Some parents think that you should never physically punish a child.)
I believe in vigorous exercise every day.
(= I believe people should get vigorous exercise every day.)