I always used to take cipro (ciprofloxacin) for recurrent UTI but became resistant to it. This time I was prescribed bactrim. Will it work this time?

A good chance... Like any other antibiotic, frequent use will create resistance and become ineffective. If you have not used that before, the chance of it working is good. If not, there are still other options. Work with your doc to get some info on preventive measures to minimize recurrence and reduce need for antibiotic use. Good luck.
Likely. Common to cycle antibiotics to reduce risk of this, and alternate antibiotic should work.