Answer
$x=\frac{\pi}{2}+2n\pi$
$x=\frac{5\pi}{6}+2n\pi$
$x=\frac{7\pi}{6}+2n\pi$
where $n$ is an integer.
Work Step by Step
$cos~2x+sin~x=0$
$cos^2x-sin^2x+sin~x=0$
$1-sin^2x-sin^2x+sin~x=0$
$(1+sin~x)(1-sin~x)+sin~x(1-sin~x)=0$
$(1-sin~x)(1+2~sin~x)=0$
$sin~x=1$ and $sin~x=-\frac{1}{2}$
The solutions in $[0,2π)$ are:
$x=\frac{\pi}{2}$
$x=\frac{5\pi}{6}$
$x=\frac{7\pi}{6}$
General solutions:
$x=\frac{\pi}{2}+2n\pi$
$x=\frac{5\pi}{6}+2n\pi$
$x=\frac{7\pi}{6}+2n\pi$
where $n$ is an integer.
