Answer
$x=\{\frac{7\pi}{6},\frac{11\pi}{6}\}$
Work Step by Step
$4cos^2(x)-4sin(x)-5=0$
$4-4sin^2(x)-4sin(x)-5=0\;\;\;\;\;\;\;\;\;\;$
$4sin^2(x)+4sin(x)+1=0\;\;\;\;\;\;\;\;\;\;\;\;$
$cos(x)=\frac{-(4)\pm \sqrt{(4)^2-(4.4.(1))}}{2.4}=\frac{-1}{2}$
$sin(x)=\frac{-1}{2}$
$x=sin^{-1}(\frac{-1}{2})$
We know $ sin(x) $ is negative in quadrant $III$ and quadrant $IV$
$x=\pi +\frac{\pi}{6}\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;x=2\pi-\frac{\pi}{6}$
$x=\frac{7\pi}{6}\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;x=\frac{11\pi}{6}$
$x=\{\frac{7\pi}{6},\frac{11\pi}{6}\}$
