The value of si$n^2$1° + si$n^2$2° + si$n^2$3° + ⋯ + si$n^2$89° is:

Pair the term for $k$° with the term for $(90-k)$°: si$n^2$$k$° + si$n^2$(90° − $k$°) = si$n^2$$k$° + co$s^2$$k$° = 1. There are 44 such pairs (1° with 89°, up to 44° with 46°), and the middle term si$n^2$45° = 1/2 is left over. Total = 44 + 1/2 = 89/2.