Calculate sin^2 1+sin^2 2 +...+ sin^2 90.

Let

(1) S = sin^2(1) +
sin^2(2)+ ...+sin^2(88) + sin^2(89)

we know sin(k) =
cos(90-k) for k = 1, 2, ..., 89, so we have

S = cos^2(89) +
cos^2(88) + ... + cos^2(2) + cos^1(1).

reverse the order of
summation, we have

(2) S = cos^1(1) + cos^2(2) + ... +
cos^2(88) + cos^2(89)

Add (1) and (2)
together;

2S = (sin^2(1) +
cos^2(1)) + (sin^2(2) + cos^2(2)) + ...+ (sin^2(89) +
cos^2(89))

= 1 + 1+ ... + 1    (the total is
89)

=
89

so devide both sides by 2,
S = 89/2 = 45.5.