$\displaystyle \cot \left( {\frac{x}{2}} \right)-2\cos \left( {\frac{x}{2}} \right)=0$

Convert cotangent into cosine divided by sine.
$\displaystyle \frac{{\cos \left( {\frac{x}{2}} \right)}}{{\sin \left( {\frac{x}{2}} \right)}}-2\cos \left( {\frac{x}{2}} \right)=0$

Factorize cosine half-angle.

$\displaystyle \cos \left( {\frac{x}{2}} \right)\left[ {\frac{1}{{\sin \left( {\frac{x}{2}} \right)}}-2} \right]=0$

Solve for x by finding all angle between but not include 0 degrees and 360 degrees.

$\displaystyle \cos \left( {\frac{x}{2}} \right)=0\ \$

$\displaystyle \frac{x}{2}=90{}^\circ ,\ 270{}^\circ$

$\displaystyle x=180{}^\circ ,360{}^\circ (\text{reject})$

$\displaystyle \frac{1}{{\sin \left( {\frac{x}{2}} \right)}}-2\ =\ 0$

$\displaystyle \frac{1}{{\sin \left( {\frac{x}{2}} \right)}}\ =\ 2$

$\displaystyle \sin \left( {\frac{x}{2}} \right)=\frac{1}{2}$

$\displaystyle \frac{x}{2}=30{}^\circ ,\ 150{}^\circ$

$\displaystyle x=60{}^\circ ,\ 300{}^\circ$

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