From Förberedande kurs i matematik 1

The angle \displaystyle 2\pi corresponds to a whole revolution and therefore we see that if we draw in a line with angle \displaystyle 2\pi relative to the positive x-axis, we will get the positive x-axis.

Because \displaystyle \cos 2\pi is the x-coordinate for the point of intersection between the line with angle \displaystyle 2\pi and the unit circle, we can see directly that \displaystyle \cos 2\pi = 1\,.