Tack has several meanings, but for the purposes of this expression we'e sticking with the nautical sense.
Sailing ships use ropes to set their sails against the wind at particular angles. Those ropes are known as tacks, in that they hold the sails the way thumbtacks hold a piece of paper on a bulletin board. As a result, the direction the ship takes relative to the wind is also known as a tack. Sailboats tack back and forth when sailing against the wind. So to take a different tack means to reposition your sails and alter course, either literally or metaphorically.
These examples show how the expression works as a synonym for "change direction" or "try a different angle:"
She resubmitted the same proposal the next year to the same reviewers, but with a more confident and transparent approach: she was straightforward about her desire to take a different tack from the type of research that had been tried before. (Nature)
None criticized her choice, but all said they would take a different tack. (New York Times)
Because they sound similar, the confusion between tack and tact is easy to understand. But tact is a synonym for discretion, not direction. You wouldn't use the article a before uncountable nouns like discretion, finesse, diplomacy, or politeness — all synonyms for tact — and you can't put an article before tact either. Tact simply is what it is: you can't use a different tact.
Despite the fact that a different tact doesn't make any sense, you'll still come across it occasionally in print:
While some state and local governments are selling office buildings and other properties to deal with the downturn, others are taking a different tact. (Wall Street Journal)
The Astros will take a different tact in their first game this series. (Seattle Times)
Don't make this tacky mistake! If you're unsure of the difference, substitute discretion, prudence, or finesse for tact and see how silly that sounds. Keep your language on course by using tack correctly.