How do you deal with a difficult coworker?

These cases are the most difficult ones. If you ask what is going on, the other person probably wouldn't tell the truth anyway. It might be a case of so called professional jealousy, which then ends up with devious tricks sabotaging your work or malicious gossip behind your back. Not a very good situation on the long run. I've been through something like this myself. I got hired as the motor pool mechanic, replacing one of the drivers who was placed in the position (but couldn't do the job) to replace the mechanic who had retired. He hated me from day one, because he thought I stole his job, as I found out later. If it wouldn't have been me then it would have been somebody else. There is usually nothing that can be done to clear such a situation, even management tried to solve this crisis by getting us all together, asking everybody what is going on, everybody denied that there was anything going on. Not even an hour later I came into the shop and the drivers were standing in a corner debating with each other, they didn't hear me coming in but I clearly heard one say: "We got to get rid of this guy, he can do everything himself, he doesn't need us". Well so what, it wasn't their job in the first place. On the long run this costs too many nerves, after 3 years I walked away from this nonsense.

Get two lunches (a kind you know she likes) and bring them in. Let her know that you brought lunch & you'd like to have lunch with her. When you're sitting down, approach it politely & say something along the lines of, "we used to make small talk while we were working. I miss that & was wondering if I did anything that irritated you that changed it." If you put it like that, she will be less likely to be defensive & more likely to tell you what's eating her. It could be nothing you've done & just something that's going on in her personal life. You might have done or said something that didn't sit well with her and you didn't even realize it. It might be (I had a situation like this) that someone else at work is filling her head with lies about something you said or did & she believes it's true. When she tells you what's up, don't get defensive, but work to fix it. Even if it was unintentional or if she's just being way oversensitive...don't be defensive & say something like, "wow, I didn't realize that happened/I did/said that/whatever. I'm sorry that you were hurt by that." Saying you're sorry she was hurt still gives an apology, but allows you to not take sole responsibility for what's going on. If you approach it like this, you might mend whatever fence is broken & have a great relationship.

The best policy is to give her some distance. She will get over it or she won't, at this point it's out of your control. If it becomes a problem you may want to have a quick chat with your boss. I would give it alot of time before I did that if I were you though.