$\text{ The sum of the maximum and the minimum }$$\text{ values of }{\sin }^2\left(\frac{2\pi }{3}+\theta \right)+{\sin }^2\left(\frac{2\pi }{3}-\theta \right)\text{ is }$

$\begin{array}{rcl} & & {\sin }^2\left(\frac{2\pi }{3}+\theta \right)+{\sin }^2\left(\frac{2\pi }{3}-\theta \right)\\ & & \end{array}$

$\begin{array}{rcl} & =& 1-\left({\cos }^2\left(\frac{2\pi }{3}+\theta \right)-{\sin }^2\left(\frac{2\pi }{3}-\theta \right)\right)\\ & =& 1-\cos \frac{4\mathrm{\pi }}{3}\cos 2\theta \\ & =& 1+\frac{1}{2}\cos 2\theta \end{array}$

$\mathrm{Minimum} \mathrm{value}=1-\frac{1}{2}=\frac{1}{2}$

$\mathrm{Maximum} \mathrm{value}=1+\frac{1}{2}=\frac{3}{2}$