Unfortunately, shaving your armpits won't make you sweat less because the practice doesn't affect the glands that produce perspiration.
The apocrine glands that produce sweat are under the skin and not in the hair itself. These glands will continue to produce perspiration even when the hair is shaved down to skin level. The perspiration will continue to leak out of the hair's openings, even if there are fewer openings because the hair is shorter or shaved down to the skin [source: Mayo Clinic].
However, shaving armpit hair can help reduce body odor [source: Willacy]. Because hair is porous, it readily absorbs odors. If you've ever sat around a campfire or spent time in a smoke-filled bar, you'll know it doesn't take long for hair to absorb the smell. Underarm hair can trap moisture, too, creating a swampy environment that odor-causing bacteria revel in. So you may feel dryer (and therefore think you're sweating less) with shaved underarms.
If you're going to shave those pits, the best way to do it is to first expose the hairs to warm water; the heat and moisture will soften and draw out the follicle, making it easier to shave cleanly. Then, exfoliate to remove debris like deodorant or dead skin cells, and apply a shave gel or cream to lubricate and protect the skin. For underarm use, a razor -- even a disposable one -- should have a flexible, pivoting head with multiple blades.
Before you start shaving, lift your arm to touch the back of your neck. This will create a flat surface under your armpit and lessen the chance of getting nicked. And, because underarm hair sprouts willy-nilly, shave in all directions – up, down and sideways – before rinsing off the shave cream. When you're done, don't apply deodorant immediately because that can sting. Instead, allow the area to air dry first [source: SheFinds].
Removing body hair is nothing new. We've been doing it for more than 2,000 years.