To be more polite avoid expressing your disagreement directly in a verbal manner. You don't have to use expressions such as: " I disagree" ," I don't think so.. "etc You will still state your opinion,but by rather showing first respect to the others point of view as in: "I do understand what you mean.,but from my point of view" " I respect your opinion, but I see things differently ...." Then state your own opinion. This way the other won't feel offended and might be more attentive to listen to your opinion.

If it's in a conversation remain calm. Keep any anger or disrespect in check. Calmly express your disagreement, all the while respecting the other person's opinions. Use phrases such as "I am afraid that I don't agree because....., I disagree because. . ."; "The problem with that is. . ."; "The way I see it. . ."; "I'm against it because. . ."; "Instead, I think that. It 's different if you are writing an essay or a paragraph You would then use more formal language.

maybe you can use this: "I understand why you think like that, but in my opinion ..... "

This is how I usually express my disagreement on a particular matter: "I beg to disagree..." or "I respect your opinion, but I can't agree..." and would usually note my reasons after. But if I'm talking to my friends, I would just say, "I don't agree..." or "I don't think so..."