lf w is a complex cube root of unity, then value of expression $\cos \left[\left\{\left(1-\mathrm{w}\right)\left(1-{\mathrm{w}}^2\right)+.......+\left(10-\mathrm{w}\right)\left(10-{\mathrm{w}}^2\right)\right\}\frac{\mathrm{\pi }}{900}\right]$

$\begin{array}{rcl} & & \cos \left(\left(1-\mathrm{w}\right) \left(1-{\mathrm{w}}^2\right)+....+\left(10-\mathrm{w}\right) \left(10-{\mathrm{w}}^2\right)\frac{\mathrm{\pi }}{900}\right)\end{array}$

$\begin{array}{rcl} & & \cos \sum _{\mathrm{r}=1}^{10} \left(\left(\mathrm{r}-\mathrm{w}\right) \left(\mathrm{r}-{\mathrm{w}}^2\right)\right) \frac{\mathrm{\pi }}{900}\end{array}$

$\begin{array}{rcl} & & \cos \sum _{\mathrm{r}=1}^{10} \left({\mathrm{r}}^2-\mathrm{r}(\mathrm{w}+{\mathrm{w}}^2)+{\mathrm{w}}^3\right)\frac{\mathrm{\pi }}{900}\end{array}$

$\begin{array}{rcl} & =& \cos \sum _{\mathrm{r}=1}^{10} ({\mathrm{r}}^2+\mathrm{r}+1) \frac{\mathrm{\pi }}{900}\end{array}$

$\begin{array}{rcl} & & \cos \left(\frac{10\times 11\times 21}{6}+\frac{10\left(11\right)}{2}+10\right) \frac{\mathrm{\pi }}{900}\end{array}$

$\begin{array}{rcl}\cos \frac{450\mathrm{\pi }}{900}& =& \cos \frac{\mathrm{\pi }}{2}=0\end{array}$