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

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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