[tex]\bf y=cot\left(x-\frac{2\pi }{3} \right)\implies y=\cfrac{cos\left(x-\frac{2\pi }{3} \right)}{sin\left(x-\frac{2\pi }{3} \right)}[/tex]
now, if the denominator turns to 0, the fraction becomes undefined, and you get a "vertical asymptote" when that happens, so let's check when is that
[tex]\bf sin\left(x-\frac{2\pi }{3} \right)=0\implies sin^{-1}\left[ sin\left(x-\frac{2\pi }{3} \right) \right]=sin^{-1}(0)
\\\\\\
x-\frac{2\pi }{3}=
\begin{cases}
0\\
\pi
\end{cases}\implies \measuredangle x=
\begin{cases}
\frac{2\pi }{3}\\
\frac{5\pi }{3}
\end{cases}[/tex]
now, at those angles, the function is asymptotic, check the picture below
