In order to prove your ex is cohabiting with a new partner in order to have your support obligation terminated or reduced under Florida law, you must prove your ex is in a supportive relationship. This begins with proving the person is cohabiting with a person to whom they are not related. This is much more difficult than it might seem.
Rarely, will the ex who is receiving alimony admit to being in a cohabitation relationship. You must prove this on your own.
Almost all legal professionals agree that the most efficient way to gather evidence to support termination of alimony is to hire a private investigator. There are some things an investigator can do for you that you cannot, or at least should not, try to do yourself.