Answer
$x=\{\frac{7\pi}{6},\frac{11\pi}{6}\}$
Work Step by Step
$2cos(x)+tan(x)-sec(x)=0$
$2cos(x)+\frac{sin(x)}{cos(x)}-\frac{1}{cos(x)}=0\;\;\;\;\;\;\;\;\;\;$ multiply each side by $cos(x)$
$2cos^2(x)+sin(x)-1=0\;\;\;\;\;\;\;\;\;\;\;\;$
$2-2sin^2(x)+sin(x)-1=0$
$2sin^2(x)-sin(x)-1=0$
$sin(x)=\frac{-(-1)\pm \sqrt{(-1)^2-(4.2.(-1))}}{2.2}=1,\frac{-1}{2}$
$sin(x)=1$
$x= sin^{-1}(1)$
$x=\frac{\pi}{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Reduce
$sin(x)=\frac{-1}{2}$
$x=sin^{-1}(\frac{-1}{2})$
We know $ sin(x) $ is negative in quadrant $III$ and quadrant $IV$
$x=\pi +\frac{\pi}{6}\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;x=2\pi-\frac{\pi}{6}$
$x=\frac{7\pi}{6}\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;x=\frac{11\pi}{6}$
$x=\{\frac{7\pi}{6},\frac{11\pi}{6}\}$
