4100.12 – There's A Cosine In It

What is the decimal value of

$\cos ^2 (1) + \cos ^2 (2) + \cos ^2 (3) + \ldots \cos ^2 (88) + \cos ^2 (89) + \cos ^2 (90).$

All angles are in degrees.

Solution

Regroup in pairs, then use complementary angles:

\begin{aligned} && \cos ^2 (1) + \cos ^2 (2) + \cos ^2 (3) + &\ldots& \cos ^2 (88) + \cos ^2 (89) + \cos ^2 (90) \\ &=& (\cos ^2 (1) + \cos ^2 (89)) + (\cos ^2 (2)+ \cos ^2 (88)) + &\ldots& (\cos ^2 (44) + \cos ^2 (46)) + \cos ^2 (45) +\cos ^2 (90)\\ &=& (\cos ^2 (1) + \sin ^2 (1)) + (\cos ^2 (2)+ \sin ^2 (2)) + &\ldots& (\cos ^2 (44) + \sin ^2 (44)) + \left(\frac{\sqrt{2}}{2} \right)^2 +0^2\\ &=& 1 + 1 + 1 + &\ldots& + 1 + \frac{1}{2} \\ &=& 44 + 1/2. \end{aligned}