If it appears to be taxed until the end of December it must have been taxed during January. i.e. before you bought it. So that tax was automatically cancelled when the vehicle changed hands. You were then obliged to tax the vehicle yourself, using the 'new keeper's details' slip from the V5C, as soon as you bought it.
You cannot now buy a vehicle with tax already on. It is cancelled when sold to you. You should have been given a change of owner slip from the V5 to get the tax for you to be legally on the road.