Ain't these 11th roots of unity ?

take $-i$ common, the expression becomes

$-i (\displaystyle \sum_{k=1}^{10} \cos(\frac{2k \pi}{11})+i \sin(\frac{2k \pi}{11}))$

$-i \displaystyle \sum_{k=1}^{10} e^{i \frac{2 k \pi}{11}}$ ..... $(1)$

11 roots of unity $1,x_{1},x_{2},x_{3}......x_{10}$

Also $1+x_{1}+x_{2}+....x_{10}=0$

$x_{1}+x_{2}+....x_{10}=-1$

in $(1)$

$-i (x_{1}+x_{2}+x_{3}+.....x_{10})$

$\boxed{i}$

Check the solution in image.