If  $\alpha ,\beta ,\gamma ,\delta \in [-\frac{\pi }{2},\frac{\pi }{2}]$ such that  $\sin \alpha +\sin \beta +\sin \gamma +\sin \delta =1$ and $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +\cos 2\delta$$\ge 10/3$    then range of  $\alpha +\beta +\gamma +\delta$  is

$\sin \beta +\sin \gamma +\sin \delta =1-\sin \alpha$

$\cos 2\alpha +1-2{\sin }^2\beta +1-2{\sin }^2\gamma +1-2{\sin }^2\delta \ge 10/3$

$\Rightarrow \cos 2\alpha \ge \frac{1}{3}+2({\sin }^2\gamma +{\sin }^2\beta +{\sin }^2\delta )$

$\Rightarrow 1-2{\sin }^2\alpha \ge \frac{1}{3}+2\frac{{(\sin \gamma +\sin \beta +\sin \delta )}^2}{3}=\frac{1+2{(1-\sin \alpha )}^2}{3}$

$\Rightarrow 3-6{\sin }^2\alpha \ge 2{\sin }^2\alpha -4\sin \alpha +3.$

$\Rightarrow 8{\sin }^2\alpha -4\sin \alpha \le 0\Rightarrow \alpha \in [0,\frac{\pi }{6}]$