Question 457926
We can use the sum/difference formulas for cosine to simplify the expression to

*[tex \LARGE \frac{ \sum_{k=0}^{20} \cos{\frac{k\pi}{20}} \cos{\frac{\pi}{4}} + \sin{\frac{k\pi}{20}} \sin{\frac{\pi}{4}}  } {\sum_{k=0}^{20} \sin(\frac{k\pi}{20})} = \sqrt{2} + \sqrt{2}\frac{ \sum_{k=0}^{20} \cos{\frac{k\pi}{20}}}{\sum_{k=0}^{20} \sin{\frac{k\pi}{20}}]

Note that the expression *[tex \sum_{k=0}^{20} \cos{\frac{k\pi}{20}] is equal to zero, because if you were to draw all the angles on a unit circle, you would see that there are corresponding pairs of angles that add up to zero, since cos x = -cos (pi - x). The denominator of the fraction is some positive number because the sine of each angle is either positive or zero. Therefore the entire expression is equal to *[tex \sqrt{2}].