The sum of solutions of the equation

$\frac{\cos x}{1+\sin x}=|\tan 2 x|$

$\Rightarrow \frac{\cos ^{2} x / 2-\sin ^{2} x / 2}{(\cos x / 2+\sin x / 2}=1 \tan 2 x$

$\Rightarrow \tan ^{2}\left(\frac{\pi}{4}-\frac{x}{2}\right)=\tan ^{2} 2 x$

$\Rightarrow 2 \mathrm{x}=\mathrm{n} \pi \pm\left(\frac{\pi}{4}-\frac{\mathrm{X}}{2}\right)$

$\Rightarrow x=\frac{-3 \pi}{10}, \frac{-\pi}{6}, \frac{\pi}{10}$

or sum $=\frac{-11 \pi}{6}$.