Without question, one obvious solution to dealing with a difficult person is to get rid of him. But that is not always easy, even if the person is your subordinate. Having institutional power over someone does not give you unlimited rights to terminate his employment, even if that is the best option. Moreover, terminating an employee can be a complex legal process.
The cost of replacing a person is another reason to try to salvage the relationship first. The person may have the talent and knowledge that are valuable to the company. If informal strategies have not been effective, it's preferable to explore more formalize ways to solve the problem. Especially if you are dealing with a difficult subordinate, you have choices you can make before you set the wheels in motion for termination. Perhaps the most powerful of these choices is to confront the need for increased accountability and diminished conflict sooner rather than later.