I think in order to be polite you must accept the possibility that you are both wrong.
There are four possibilities:
- You have correctly identified areas where he is wrong, and you are correct. i.e. He is wrong and you are right.
- You have correctly identified areas where he is wrong, but you are also wrong. i.e. you are both wrong.
- Your manager is correct, and you have misinterpreted what he meant, you have proposed an equally valid solution. i.e. you are both correct.
- Your manager is correct, and you have misinterpreted what he meant, you have proposed an incorrect solution. i.e. he is right and you are wrong.
If you respect your manager's authority (or wish to at least seem to), you must assume that he (or she) believes he is correct, and that he has come to his conclusions in a logical manner. By the wording of your question you believe yourself to be correct, and I would assume you have also come to your conclusions in a logical manner.
If you both started from the same information, and both took logical routes you could both be correct.
Therefore a polite (and constructive) response would be:
- First raise that you have come to a different conclusion to your manager and that you would like to understand where the difference comes from. (Do not claim your conclusion is better than his)
- Confirm that you both have the same starting information. If you have uncovered more information that he did not have, then he may come to the same conclusion as you and thank you for informing him of this new information.
- If they have the same starting information, then you must have applied a different process to this information to end up at different answers. You can then compare how you came to your answer and how he came to his and decide between you which answer is the best.
The key to being helpful and polite is to accept the possibility that you may be wrong. If you go into the discussion convinced that you are correct and he is wrong, and you are unwilling to listen to him, then he will not listen to you.