when does dana find out terry is batman

Dana doesn't discover Terry's secret identity in the Batman Beyond animated series itself. There are, however, some hints and unresolved plotlines that suggest she might have figured it out eventually:

Clues and Suspicions: Throughout the series, there are moments where Dana expresses suspicion about Terry's disappearances and injuries. For example, she questions his alibis and notices his odd behavior after fighting crime as Batman.

Open Ending: The series finale leaves things open-ended. Terry continues as Batman, and there's no definitive answer regarding whether Dana ever learns the truth.