Question

Find the value $\sum _{k=1}^6(\sin \frac{2\pi k}{7}-\cos \frac{2\pi k}{7})$

Answer 1

$\sum _{k=1}^6\left(\sin \frac{2\pi k}{7}\right)-\sum _{k=1}^6\left(\cos \frac{2\pi k}{7}\right)$
$=\sum _{k=0}^6\sin \frac{2\pi k}{7}-\sum _{k=0}^6\cos \frac{2\pi k}{7}+1$
$={\sum }_{k=0}^6$(Sum of imaginary part of seven seventh roots of unity)-${\sum }_{k=0}^6$(Sum of real part of seven seventh roots of unity)+1
$=0-0+1=1$