The old fans or fronds that you cut off will not grow back. A Mexican Fan Palm like most palms only has one growing point, the top of the trunk. That is where new fronds are formed as the trunk grows slowly taller and taller. When fronds start to die back near the base of the tree's bushy top you can cut them off. Just don't prune off too much of the top. When you are done, the lowest remaining fronds should be drooping downward slightly. Don't prune lower ones until the remaining fronts are more upright than horizontal.
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