How do we know that the Sun rotates? By watching sunspots from day to day.
In these two photos, taken about a day apart, you can see a new sunspot cluster rotating into view at the lower left. There's also an old cluster rotating out of view at the upper right.
The Sun rotates approximately once a month. In fact, the Sun is not solid, and the period of rotation is actually a few days faster at the equator, and a few days slower at the poles.
Sunspots do form, change shape and size, and disappear over time, but these changes happen slowly enough that the rotation of the Sun can be observed by watching the sunspot patterns seem to move across the disk as the Sun rotates.
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