[tex]\displaystyle\\
\sqrt{3}~ \text{ctg}\left(x-\frac{\pi}{2}\right) +3=0\\\\
\sqrt{3}~ \text{ctg}\left(x-\frac{\pi}{2}\right) =-3\\\\
\text{ctg}\left(x-\frac{\pi}{2}\right) =-\frac{3}{\sqrt{3}}\\\\
\text{ctg}\left[-\left(\frac{\pi}{2}-x\right)\right] =-\frac{3}{\sqrt{3}}\\\\
\text{-ctg}\left(\frac{\pi}{2}-x\right)=-\frac{3}{\sqrt{3}}\\\\
\text{ctg}\left(\frac{\pi}{2}-x\right)=\frac{3}{\sqrt{3}}\\\\
\text{tg }x=\frac{3}{\sqrt{3}}\\\\
\text{tg }x=\sqrt{3}\\\\
x=\boxed{\frac{\pi}{3} +k\pi}[/tex]
