What a wonderful question you've asked. Believe is a verb, meaning accept that something is true, especially without proof.it is however possible to believe someone but not trust them until they earn your trust. So trust is an act of reliance based on accepting something to be true.
Also, when you fully believe in a person's words or actions, you generally place your faith in them. Believe is used to show one's acceptance to another words. Trust is used to demonstrate reliance on the other's overall character. Both of the statement, I don't believe him and I don't trust him are correct, depending on the degree of faith you are having or giving to him.
Do in a simple term, you trust someone or something meaning that you rely on them and want to believe in ther