Given $\cos x=\frac{2 \cos y-1}{2-\cos y}$

Applying compoundo and dividendo rule

$\frac{1+\cos x}{1-\cos x}=\frac{2 \cos y-1+2-\cos y}{2-\cos y-2 \cos y+1}$

$\Longrightarrow \frac{2 \cos ^{2} x / 2}{2 \sin ^{2} x / 2}=\frac{1+\cos y}{3(1-\cos y)}=\frac{\cos ^{2} y / 2}{3\left(2 \sin ^{2} y / 2\right)}$

$\Longrightarrow \tan ^{2} \frac{x}{2}=3 \tan ^{2} \frac{y}{2}$

$\Longrightarrow \tan \frac{x}{2} \cot \frac{y}{2}=\sqrt{3}$