$1-2{\cos}^22x-\sin x=0$
$\Rightarrow-(\cos 4x)-\sin x=0$
$\Rightarrow\cos 4x=-\sin x=\sin(-x)=\cos(\dfrac{\pi}{2}+x)$
$\Rightarrow 4x=\pm(\dfrac{\pi}{2}+x)+k2\pi$
$\Rightarrow \left[ \begin{array}{l} x=\dfrac{\pi}{6}+k\dfrac{2\pi}{3} \\ x=\dfrac{-\pi}{10}+k\dfrac{2\pi}{5}\end{array} \right .$
+) $0\le\dfrac{\pi}{6}+k\dfrac{2\pi}{3} \le3\pi$
$\Rightarrow \dfrac{-1}{4}\le k\le\dfrac{17}{4}$
$\Rightarrow k=\{0;1;2;3;4\}$
$\Rightarrow x=\{\dfrac{\pi}{6};\dfrac{5\pi}{6};\dfrac{3\pi}{2};..\dfrac{17\pi}{6}\}$
+) $0\le\dfrac{-\pi}{10}+k\dfrac{2\pi}{5} \le3\pi$
$\Rightarrow 0,25\le k\le7,75$
$\Rightarrow k=\{1;2;3;4;5;6;7\}$
$\Rightarrow x=\{\dfrac{3\pi}{10};\dfrac{7\pi}{10};..\dfrac{27\pi}{10}\}$
$\Rightarrow \sum=18\pi$
