The concept of a broad-brimmed hat with a high crown worn by a rider on horseback can be seen as far back as the Mongolian horsemen of the 13th century. The hat has a tall crown that provides insulation, and a wide brim that provides shade.

from: https://en.wikipedia.org/wiki/Cowboy_hat

The HatteristThe Hatterist

The American West is symbolized by the trailblazing, sharpshooting, horseback-riding cowboy of American lore. Portrayed with the ubiquitous outfit of denim jeans with fringed chaps, boots with spurs & a plaid shirt. And though African-American cowboys don’t play a part in the popular narrative, historians estimate that one in four cowboys were black. This & other common tales about the cowboys of Hollywood legend aren't true:

1. They rarely fought Indians.
2. They were more likely to die from falling off a horse than in a gunfight.
3. They didn't wear those tall, wide-brimmed cowboy hats you see in Westerns.

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nationalcowboymuseum.org
https://www.youtube.com/watch?v=HNqAzyBfXe0