From Förberedande kurs i matematik 1

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

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