Question

The value of $\displaystyle\sum_{ r =0}^{10} \cos ^{3} \frac{ r \pi}{3}$ is equal to $\frac{- a }{ b }$ then the value of $b$ is (where g.c.d of $(a, b)$ is 1$)$

Answer 1

$\displaystyle\sum_{ r =0}^{10} \cos ^{3} \frac{ r \pi}{3}=\frac{1}{4} \displaystyle\sum_{ r =0}^{10}\left(3 \cos \frac{ r \pi}{3}+\cos r \pi\right)$ $=\frac{1}{4}\left[3\left(\cos 0+\cos \frac{\pi}{3}+\ldots .+\cos \frac{10 \pi}{3} +(1-1+\ldots . .-1=1)\right]\right.$