If w is a complex cube root of unity then the value of $\left[\cos \left\{\left(1-\mathrm{w}\right)\left(1-{\mathrm{w}}^2\right)+\left(2-\mathrm{w}\right)\left(2-{\mathrm{w}}^2\right)+\dots \dots +\left(10-\mathrm{w}\right)\left(10-{\mathrm{w}}^2\right)\right\}\frac{\mathrm{\pi }}{9}\right]$is

$\begin{array}{rcl} & & \cos (1-\mathrm{w}) \left(1-{\mathrm{w}}^2\right)+(2-\mathrm{w}) \left(2-{\mathrm{w}}^2\right)...+(10-\mathrm{w}) \left(10-{\mathrm{w}}^2\right) \frac{\mathrm{\pi }}{9}\\ & & \cos \sum _{\mathrm{r}=1}^{10} \left({\mathrm{r}}^2-\mathrm{r}(\mathrm{w}+{\mathrm{w}}^2)+{\mathrm{w}}^3\right) \frac{\mathrm{\pi }}{9}\\ & & \cos \sum _{\mathrm{r}=1}^{10} \left({\mathrm{r}}^2+\mathrm{r}+1\right) \frac{\mathrm{\pi }}{9}\\ & & \cos \left(\frac{10 \left(11\right) \left(21\right)}{6}+\frac{10 \left(11\right)}{2}+10\right) \frac{\mathrm{\pi }}{9}\\ & & \cos \left(385+55+10\right) \frac{\mathrm{\pi }}{9}\\ & & \cos 450\frac{\mathrm{\pi }}{9}\\ & & \cos 50\mathrm{\pi }\\ & =& 1\end{array}$