The histories of kush and egypt are intertwined with each other because they were reliant on each other. how did their interdependence work, and why did each take control of the other in turn?

The Kingdom of Kush was located in Nubia, between Egypt and Sudan.

In simple terms, Egypt relied on Nubia's gold and Nubia relied on Egypt's grain. Each had something that the other could not produce by themselves, that's why they were willing to be interdependent. So much so, that historians to this day have tried to understand the entire sociopolitical structure of Kush through egyptian models, which is somewhat of a mistake as the Nubian state  presented distinct cultural conditions since its earliest periods. We could say they were strongly reliant on each other economically, but Egypt has sort of taken control of the history of Kush.

Hope this clears things up!