$\sum _{k=1}^6\left(\sin \frac{2\pi k}{7}-i\cos \frac{2\pi k}{7}\right)$

$=\left(\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}+\sin \frac{8\pi }{7}+\sin \frac{10\pi }{7}+\sin \frac{12\pi }{7}\right)-i\left(\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{8\pi }{7}+\cos \frac{10\pi }{7}+\cos \frac{12\pi }{7}\right)$

$=\left(\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}+\sin \left(2\pi -\frac{6\pi }{7}\right)+\sin \left(2\pi -\frac{4\pi }{7}\right)+\sin \left(2\pi -\frac{2\pi }{7}\right)\right)-i\left(\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\cos \left(2\pi -\frac{6\pi }{7}\right)+\cos \left(2\pi -\frac{4\pi }{7}\right)+\cos \left(2\pi -\frac{2\pi }{7}\right)\right)$

$=\left(\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{6\pi }{7}-\sin \frac{6\pi }{7}-\sin \frac{4\pi }{7}-\sin \frac{2\pi }{7}\right)-i\left(\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{6\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{2\pi }{7}\right)$ (Using $\sin \left(2\pi -\theta \right)=-\sin \theta \&\hspace{0.17em}\cos \left(2\pi -\theta \right)=\cos \theta$)

$=-2i\left(\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}\right)$

$=-2i\frac{\sin \left({\displaystyle \frac{3\pi }{7}}\right)}{\sin \left(\frac{\pi }{7}\right)}\cos \left(\frac{2\pi }{7}+\frac{2\pi }{7}\right)$ (Using $\cos \left(a\right)+\cos \left(a+b\right)+\cos \left(a+2b\right)+....+\cos \left(a+\left(n-1\right)b\right)=\frac{\sin \left({\displaystyle \frac{nb}{2}}\right)}{\sin \left({\displaystyle \frac{b}{2}}\right)}\cos \left(a+\left(n-1\right)\frac{b}{2}\right)$)

$=-2i\frac{2\sin \left({\displaystyle \frac{3\pi }{7}}\right)\cos \left({\displaystyle \frac{4\pi }{7}}\right)}{2\sin \left(\frac{\pi }{7}\right)}$

$=-2i\frac{\sin \left({\displaystyle \frac{7\pi }{7}}\right)-\sin \left({\displaystyle \frac{\pi }{7}}\right)}{2\sin \left(\frac{\pi }{7}\right)}$ (Using $2\sin A\cos B=\sin \left(B+A\right)=\sin \left(B-A\right)$)

$=i$