4100.12 – There's A Cosine In It

Find the decimal value of

cos⁡2(1)+cos⁡2(2)+…+cos⁡2(89)+cos⁡2(90)\cos ^2 (1) + \cos ^2 (2) + \ldots + \cos ^2 (89) + \cos ^2 (90)

All angles are in degrees.

Solution

Regroup in pairs, then use complementary angles:

cos⁡2(1)+…+cos⁡2(90)=(cos⁡2(1)+cos⁡2(89))+(cos⁡2(2)+cos⁡2(88))⋮+(cos⁡2(44)+cos⁡2(46))+cos⁡2(45)+cos⁡2(90)=(cos⁡2(1)+sin⁡2(1))+(cos⁡2(2)+sin⁡2(2))⋮+(cos⁡2(44)+sin⁡2(44))+(22)2+02=1+1+1+…+1+12=44+1/2\begin{aligned}\cos ^2 (1) + \ldots + \cos ^2 (90) &= (\cos ^2 (1) + \cos ^2 (89)) \\[0.5em]& \quad + (\cos ^2 (2)+ \cos ^2 (88)) \\[0.5em]& \qquad \vdots \\[0.5em]& \quad + (\cos ^2 (44) + \cos ^2 (46)) \\[0.5em]& \quad + \cos ^2 (45) +\cos ^2 (90) \\[0.5em]&= (\cos ^2 (1) + \sin ^2 (1)) \\[0.5em]& \quad + (\cos ^2 (2)+ \sin ^2 (2)) \\[0.5em]& \qquad \vdots \\[0.5em]& \quad + (\cos ^2 (44) + \sin ^2 (44)) \\[0.5em]& \quad + \left(\dfrac{\sqrt{2}}{2} \right)^2 +0^2 \\[0.5em]&= 1 + 1 + 1 + \ldots + 1 + \dfrac{1}{2} \\[0.5em]&= 44 + 1/2\end{aligned}