Countries in Western Europe, especially the smaller ones, are quite used to the idea of accepting next door's currency. I don't actually think you'd have much of a problem spending Euros in Copenhagen - Danes are used to cross-border visitors (especially from Germany) spending their own currency. (I grew up in Northern Germany, not far from the Danish border)

As others have said, you are likely to pay more if you pay in Euros, and any change will probably be in Kroner, so you will end up with pocketfuls of local coins.

Most places in Copenhagen accept credit cards, BTW (including cabs)