Trig

Use the fact that:

sin(x) = cos(90-x)

We then can rewrite our equation as:

cos²(0) + cos²(1) + cos²(2)........ cos²(44) + sin²(45) + sin²(44) ........ sin²(0).

Next we make use of a basic trig identity:

$\cos^2\theta + \sin^2\theta = 1$

Given that the two angles are the same.

We realize we have 0-44 = 45 total such pairs in our rewritten equation(can you see how?).

This gives us 45 + sin²(45) =  $45+(\sqrt2/2)^2 = 45 + 2/4 = 45 + 1/2 = 45.5$

Does any one get 44.5 ?    I used an online  summation calculator.....