Answer
$x=\frac{5\pi}{3}+2n\pi$
$x=\pi+2n\pi$
where $n$ is an integer.
Work Step by Step
$sin\frac{x}{2}+cos~x=0$
$sin\frac{x}{2}=-cos~x$
$\pm\sqrt {\frac{1-cos~x}{2}}=-cos~x~~$ (square both sides)
$\frac{1-cos~x}{2}=cos^2x$
$2~cos^2x+cos~x-1=0$
$2~cos^2x+2~cos~x-cos~x-1=0$
$2~cos~x(cos~x+1)-(cos~x+1)=0$
$(2~cos~x-1)(cos~x+1)=0$
$cos~x=\frac{1}{2}$ or $cos~x=-1$
$cos~x=\frac{1}{2}$:
$x=\frac{\pi}{3}$ or
$x=\frac{5\pi}{3}$
Testing the solutions:
$sin\frac{\frac{\pi}{3}}{2}+cos~\frac{\pi}{3}=\frac{1}{2}+\frac{1}{2}\ne0$
$sin\frac{\frac{5\pi}{3}}{2}+cos~\frac{5\pi}{3}=-\frac{1}{2}+\frac{1}{2}=0$
$cos~x=-1$:
$x=\pi$
Testing the solution:
$sin\frac{\pi}{2}+cos~\pi=1-1=0$
The period of $cos~x$ is $2\pi$. The general solutions are:
$x=\frac{5\pi}{3}+2n\pi$
$x=\pi+2n\pi$
where $b$ is an integer.
