"They are both tired."
Translation:De er begge trætte.
Både goes before lists ("Jeg kan lide både æg og ost" = "I like both eggs and cheese") but "begge" just refers to two things that have already been mentioned. For example:
A: Kan du lide æg eller ost? (Do you like eggs or cheese?)
B: Jeg kan lide dem begge (I like both of them)
N.B. Både isn't restricted to just two things, whereas "begge" is