Osu is actually not a word in Japanese at all. There is some disagreement of the roots words, but we know for certain osu is a contraction. The first theory is that osu is the shortened version of "Oshi Shinobu," which in English translates as "to endure under pressure." The second is that it actually comes from "Onegaishimasu," commonly translated as "Please teach me." So why did we shorten these words to osu, and how did it become the go-to phrase in nearly all Western martial arts schools?
So if it's not even a word, is it appropriate to use in the dojo and if so, when? First and foremost, if you are a student of an instructor who insists on using "osu," you should follow their example. It is their school or system after all, and would be disrespectful not to. However, in my opinion, the only time it is appropriate is when bowing to someone of lower rank or position, using it as the shorthand version of "onegaishimasu" which the lower rank should be saying to you completely, or when bowing to the shomen, kamiza or kamidana. For most other situations in which "osu" is commonly used, such as to show understanding of the instruction, the phrase "Hai, wakarimasu" or simply "Hai" would be more appropriate and respectful.
Hopefully this article has helped you in your understanding of proper Japanese etiquette. Stay tuned for more articles on the subject!