"Trust" is belief with a high confidence level in the reliability, truth, or ability.
Trust is based on people knowledge about another.
If you trust someone which means you Believe what they said, behaviors, abilities are true, and you rely on them.
If someone gives you an information and you accept it without proof, it's means you trust that person.
Trust是信任的意思。表示对某事某物很高的信心。这种信任来自理性的认可。如果你信任一个人，你会在没有任何证据的前提下，选择相信这个人所做的一切。
Examples:
I trust all products of that company because of very good reputation over 100 years.
我信任这家公司的产品，因为这家公司有非常好的名声。
Good relationship based on mutual (彼此的) trust.
好的关系需要彼此的信任。
I trust you (I have confidence in you)
我信任你。
"Believe" means to accept something as true. Believe is a value of acceptance to facts, and it is based on thoughts and observations.
If someone gives you an information and that the information is true finally, then you will believe that person.
Believe是基于个人的思考与观察从而相信、接受的意思。引申开来有信仰的意思。
Example:
I believe you (I think what you said is true)
我相信你(我认为你说的是真的)。
In addition, "believe in" can be used to refer person confidence in another person. This phrase has similar meaning with “trust”.
"believe in" 是信任的意思，与trust意思相似。
Example:
I believe in you (I have confidence in you).
我信任你(我对你有信心，愿意相信你)。
