It's the instrumental case. However, I'm not sure why the adjective has been declinated, as it is not paired with a noun e.g. it would be Jestem zmęczon/y/a, rather than jestem zmęczon/ym/ą. This can be contrasted to, for e.g., ''Jestem zmęczonym mężczyzną'', which does have a noun. Can anyone help with an answer to this? MOD? Thanks. (P.S. I can recommend the following YouTube video for anyone who needs more info re the instrumental case ... https://www.youtube.com/watch?v=bM-Uzgy225A)
A contact of mine, who is a professional interpreter of Polish/English gave me the answer to this question:-
Under normal circumstances, a stand alone adjective is NOT declinated (e.g. Jestem zmęczony), however there are exceptions to this.
1) A stand alone adjective following certain prepositions WILL be declinated e.g. Tom wygląda na szczęśliwego. = Tom looks happy.
2) A stand alone adjective having an IMPLIED physical/material form will also be declinated e.g. Nie podobała mi się czerwona, więc kupiłam zieloną. = I didn’t like the red (one), so I bought the green (one). (In this example, the omitted noun is feminine).
So, going back to the original sentence …
Nie możesz pracować będąc zmęczonym. = this would imply = ‘’You cannot work, being (a) tired (person).’’
Hope this helps. (Thanks to alukasiak for his help with this one).