The value of co$s^2$10° + co$s^2$20° + co$s^2$30° + co$s^2$40° + co$s^2$50° + co$s^2$60° + co$s^2$70° + co$s^2$80° is:

Group the angles that add to 90°: co$s^2$10° + co$s^2$80° = co$s^2$10° + si$n^2$10° = 1, and the same happens for (20°, 70°), (30°, 60°) and (40°, 50°). Four pairs, each worth 1, give 4. Arithmetic check: 0.9698 + 0.8830 + 0.7500 + 0.5868 + 0.4132 + 0.2500 + 0.1170 + 0.0302 = 4.000. Counting 1 per term gives 8, and there is no stray half here because no angle is its own partner — 45° is not in the list.