Cunning Complex 5

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

$\Rightarrow -i\sum _{ k=1 }^{ 6 }{ \left( cos\frac { 2\pi k }{ 7 } +isin\frac { 2\pi k }{ 7 } \right) } =-i\sum _{ k=1 }^{ 6 }{ \left( { e }^{ \frac { 2\pi k }{ 7 } } \right) }$ $\Rightarrow -i\sum _{ k=1 }^{ 6 }{ \left( \alpha ^{ k } \right) } \quad \quad \quad ,where\quad \alpha ={ e }^{ \frac { 2\pi }{ 7 } }$ $=-i\frac { \alpha ({ \alpha }^{ 6 }-1) }{ (\alpha -1) } =-i\frac { ({ \alpha }^{ 7 }-\alpha ) }{ (\alpha -1) }$ $=-i\frac { (1-\alpha ) }{ (\alpha -1) } \quad \quad \quad \quad \because { \alpha }^{ 7 }={ e }^{ \frac { 2\pi \times 7 }{ 7 } }={ e }^{ 2\pi }=1$ $=\boxed{i}$