Trust and Believe are two English words that can be used synonymously in some contexts and they will have the same meaning. However, there are a few differences between the two words. Trust is a word that can be used as a noun, it can be used as a verb and also as an adjective. As a noun, it means confidence in, or reliance on some person or entity. It also means, dependence on something in the future. For example: Trust is needed to actualize this contract.
As a verb, it means to have total confidence in someone, or to hope confidently or believe that something is going to happen. For example: we are trusting on God to reveal Himself to us. Believe, on the other hand, is used only as a verb. It means to have faith in something or to accept something as true without necessarily having an obvious certainty to do so. For example: I believe I'm going to be paid tomorrow.