Honestly I am not rather splendid at the whole 'explaianing the danish pronunciation' thing, but what I can do, is to give you a link with a very good articulation of the "shotened" tager. The thing is, that we actually only uses the 'full version' when we want to be distinct about the word.
Even this Danish dictionary site, which has pronuncations, says tar as present of at tage.
Here is the link. Under Vis overblik, at UDTALE, you click the third speaker (the one preceded by the words præsens: [ˈtɑˀ]