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) = \(45+(\sqrt2/2)^2 = 45 + 2/4 = 45 + 1/2 = 45.5\)
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