What is the difference between trust and believe , and when should we use them . For example "I don't believe him " is correct or "I don't trust him " .
Trust (verb) is defined as: believe in the reliability, truth, or ability of: I should never have trusted her. When you fully believe in a person's words or actions, you generally place your faith in them.
Believe (verb) is defined as: accept that (something) is true, especially without proof: the superintendent believed Lancaster's story.
It is possible to believe someone but not trust them until they earn your trust fully, therefore trust is more of an act of reliance based on accepting something to be true.
Believe is used to show one's acceptance of another's words. Trust is used to demonstrate the reliance on the other's overall character.
Both of the statements, I don't believe him and I don't trust him are correct, depending on the degree of faith you are giving to him.