How do changes in the earths tilt affect the amount of sunlight that strikes on north and south poles?

It turns out that if the Earth's axis is tilted at all, no matter how much or how little, then the north and south poles would still each have a period of six months when the sun doesn't set, and a period of six months when the sun doesn't rise. And by the way ... the degree of tilt of the Earth's axis doesn't change. At least, not enough to notice, or to have any noticeable effect on anything in our daily life.

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