It depends. If you are a citizen or resident of the United States, you do have to pay income taxes on your trading profits.
There is really no way around this. The amount of taxes you pay might vary if you trade under your own name or under the name of a business entity (consult a tax professional if you are considering setting up a business for trading purposes).
On the other hand, if you are a foreigner (non-U.S. resident) you can trade without paying any taxes to the United States government. That is one of the greatest incentives that the United States offers foreign investors.