How to deal with a complicated SIL?

I recently found out about a big lie my sil said about me. I'm pretty upset. Basically she told a cousin of hers (who is a pretty good friend of mine) that although I've let her borrow my stuff I've never given her anything pretty. All because I gave said cousin a cosmetics bag full with all sorts of makeup which sil proceeded to open and try on as if they were hers. Cousin was shocked and didn't say anything. So this really upset me. Sil is 18 now this happened when she was 17. I've been married to her brother almost 5 years now. And this is a totally false statement. I could write paragraphs listing all the things I've gifted her, including hair irons, makeup, perfumes, shoes, lots and lots of clothes, etc etc. Oh jewelry etc etc...literally thousands of dollars spent on her. I knew she was ungrateful, she never says thankyou for anything, and nothing is ever enough, but never imagined she'd make up lies like this making me look bad in front of others. She also said she didn't like me and that I was weird and all sort of stuff. I'm afraid to confront her she always turns things around and my mil believes her. I've had problems with her before because she's stolen my things and basically problems started when I wouldn't do what she wanted and stood up for myself (she expected me to wash her clothes and buy her whatever she "needed"). How would you handle the situation?