Do not try to change an arrogant person. They won’t listen, and they’re not interested in any opinion but their own. The best strategy is distance. If you have to deal with someone like this, be clear, direct, and don’t give more explanations than necessary. Don’t play their game or try to prove anything. Arrogance thrives on conflict.
If you’re forced to interact with them, don’t get emotionally involved. Listen just enough, respond calmly, and stay in control. Rudeness seeks a reaction, and the best way to win is not to give it.