The 180 numbers cos1°, cos2°, cos3°, …, cos180° are all added together. The total is:

Pair each cos k° with cos(180° − k°) = −cos k°: the pairs (1°, 179°), (2°, 178°), … , (89°, 91°) wipe each other out, using up 178 of the terms. That leaves cos90° = 0 and the unpaired cos180° = −1, so the total is −1. Assuming everything cancels gives 0, taking cos180° as +1 gives 1, and counting cos180° a second time gives −2.