$\begin{aligned} & \sum_{k=1}^6\left(\sin \frac{2 \pi k}{7}-i \cos \frac{2 \pi k}{7}\right) \\ & =-i \sum_{k=1}^6\left[\cos \left(\frac{2 \pi k}{7}\right)-\frac{1}{i} \sin \left(\frac{2 \pi k}{7}\right)\right] \\ & =-i \sum_{k=1}^6\left[\cos \left(\frac{2 \pi k}{7}\right)+i \sin \left(\frac{2 \pi k}{7}\right)\right] \\ & =-i\left[\sum_{k=1}^6 e^{\frac{i 2 \pi k}{7}}\right] \ldots \text { (i) }\left\{\because e^{i \theta}=\cos \theta+i \sin \theta\right\}\end{aligned}$
$$
\begin{array}{r}
\text { Now, }\left[1+e^{\frac{i 2 \pi}{7}}+e^{\frac{i 4 \pi}{7}}+e^{\frac{i 6 \pi}{7}}+e^{\frac{i 8 \pi}{7}}+e^{\frac{i 10 \pi}{7}}+e^{\frac{i 12 \pi}{7}}=0\right. \\
\left\{\because z^7=1 \text { then, roots } 1, e^{\frac{i 2 \pi}{7}}, e^{\frac{i 4 \pi}{7}} \ldots e^{\frac{i 12 \pi}{7}}\right\} \\
=1+\sum_{k=1}^6 e^{i(2 \pi k / 7)}=0=\sum_{k=1}^6 e^{i(2 \pi k / 7)}=-1 \quad \ldots \text { (ii) }
\end{array}
$$
From Eqs. (i) and (ii), we get $=-i(-1)=i$