I have a really close friend who is dating my other friend. She has been my friend for a long time and I don't want to lose her. She says she cannt trust me. How can I show her I really care about her, she can trust me and I will be honest with her all the time?

Are you trying to pursue a relationship with her? If so, then you can't go behind your friend's back to try to date her. This makes you look dishonest to her. If you are just good friends and you don't want a romantic relationship, then you have to build the trust up. It sounds like she doesn't want a romantic relationship right now. Just be there for her and never lie to her. Over time, she will appreciate your willingness to listen and your honesty, whether you become more than friends. Trust takes time to build. The best way to show someone you care is to be there for them to talk to no matter what.