It didn't happen with Reagan. Reagan cut taxes and the deficit disappeared the moment Clinton came into office. Even Clinton cut taxes.
The problem with the deficit is not how much tax we collect, but how much we spend. We need to prune the problemed spending, like Welfare (going to many who don't need it- don't cut benefits to those that do need it), Social Security (we have thousands of Mexican citizens collecting SS), and pork barrel spending (the govt spent 11 million to study the lifestyle of gay indians...wtf?). Our priorities for the federal government are as they have always been: security of the people and then the economy. Military cuts are the worst that can happen because military and intelligence cuts literally caused the 3000 dead on 9/11. Keep the military funds where they are (or boost them) and cut the unnecessary pork barrel projects. And most of all, cut taxes so that the economy will be stimulated. Reagan did it with superb success and Kennedy did it with pretty good success as well. No other economic model has shown any promise.