\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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