Why don't she trust me ?

she didn't trust u because ,as u told, u are not good to others and may be she thinks that u are not true to her and telling lies towards her .if she want to trust u ,he good to others and help others and u;r sister also.then she can get trust towards u.

Well, it seems like you answered your question yourself. If you speak lies to everyone else and she knows it, then how does she know you won't do it to her. She might know you love her, but loving and trusting someone are two different things. There was a time that I loved my sister, but did not trust her for the things she was doing. It took a long time to trust her completely again. If this bothers you, then you need to earn her trust by stop lying to others. Show her that you are changing. It will take a long time, but it is worth, especially to the person you love. Good Luck!