If $\omega$ is a complex cube root of unity, then $\cos \left\{\left[(1-\omega )\left(1-{\omega }^2\right)+(2-\omega )\left(2-{\omega }^2\right)+(3-\omega )\left(3-{\omega }^2\right)+(4-\omega )\left(4-{\omega }^2\right)\right.\right.$

$\left.\left.+(5-\omega )\left(5-{\omega }^2\right)\right]\frac{2\pi }{75}\right\}$ equals

$\begin{array}{r}(1-\omega )\left(1-{\omega }^2\right)=(1-\omega )(2+\omega )\left[\therefore 1+\omega =-{\omega }^2\right]\\ =2+1=3\end{array}$

Similarly, we get $(2-\omega )\left(2-{\omega }^2\right)=7$

$\begin{array}{r}(3-\omega )\left(3-{\omega }^2\right)=13\\ (4-\omega )\left(4-{\omega }^2\right)=21\\ (5-\omega )\left(5-{\omega }^2\right)=31\end{array}$

So $\cos \left\{\left[(1-\omega )\left(1-{\omega }^2\right)+\dots .+(5-\omega )\left(5-{\omega }^2\right)\right]\frac{2\pi }{75}\right\}$

$\begin{array}{l}=\cos \left\{[3+7+13+21+31]\frac{2\pi }{75}\right\}\\ =\cos \left\{2\pi \right\}=1\end{array}$