sin square 5 plus sin square 10 plus sin square 15 plus......... plus sin square 90
assuming the angles are in degrees, you apparently want to find
sin^2 5° + sin^2 10° + ... + sin^2 90°
Rearrange them in pairs, working from the ends.
sin^2 0° + sin^2 90° + sin^2 5° + sin^2 85° + ... + sin^2 45°
now recall that cos(x) = sin(90-x) so you have
(sin^2 0° + cos^2 0°) + (sin^2 5° + cos^2 5°) + ... + sin^2 45°
There are 9 pairs of angles in parentheses.
sin^2 x + cos^2 x = 1
so the sum is 1+1+1+1+1+1+1+1+1 + 1/2 = 9 1/2