Answered by Shaykh Abdurrahman ibn Yusuf Mangera
In the name of Allah the Inspirer of truth.
It is not permissible to bow in these circumstances. Although it may not have any religious significance to the art, however, as a Muslim it is an empathetically prohibited act for you. It is an act reserved for Allah alone, and doing it for other than Allah is either unlawful, or can leads one to kufr if done with intention to worship. If it had been permissible, even as to honor someone, it would have been permissible to do it for the Prophet (upon him be peace) or one's elders, which is not the case.
The Messenger (upon him be peace said) "If I were to command anyone to prostrate before anyone I would command the wife to prostrate before the husband" (Mishkat al-Masabih).
My son is in a karate class and his teachers respects the fact that as a Muslim he is not allowed to bow, and hence makes sure that he does not bow. Perform two rak'ats prayer and then go and discuss the issue with your instructor, and it is hoped that he will be understanding of your situation.
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