The sum si$n^2$1° + si$n^2$3° + si$n^2$5° + ⋯ + si$n^2$89°, taken over the 45 odd whole degrees from 1° to 89°, is:

If $k$ is odd then $90-k$ is odd too, so the list pairs with itself: si$n^2$$k$° + si$n^2$(90° − $k$°) = si$n^2$$k$° + co$s^2$$k$° = 1. The pairs are (1°, 89°), (3°, 87°), … , (43°, 47°) — 22 pairs worth 22 — and the self-paired middle term si$n^2$45° = 1/2 is left over, giving 22 + 1/2 = 45/2 = 22.5. Forgetting the middle term leaves 22; 89/2 is the total over all 89 whole degrees, and 45 would need every one of the 45 terms to be 1.