A local record is made when the registration is performed, and that will be gone or invalidated when you reinstall Windows. Impulse just does it for you in the background when you install through it.

All you need to do is enter the serial key from Impulse and the same email address you used previously, to activate the game on that machine. You should also be able to use the detect application option to get Sins back in Impulse, and right click the game after that and choose the reactivate option (assuming GameStop has not removed it).