Answer: \(x=-\frac{7\pi}{12},-\frac{\pi}{12},\frac{5\pi}{12},\frac{11\pi}{12}\).

The key point is to reduce the equation to a standard trigonometric equation, then use the given interval to list all valid solutions.

The equation is

\(\operatorname{cot}\left(2x+\frac{\pi}{3}\right)=\sqrt3.\)

Therefore

This occurs when

\(2x+\frac{\pi}{3}=\frac{\pi}{6}+k\pi,\)

where \(k\) is an integer. Hence

\(2x=-\frac{\pi}{6}+k\pi\)

and

\(x=-\frac{\pi}{12}+\frac{k\pi}{2}.\)

For \(-\pi\lt x\lt \pi\), the possible values are obtained from \(k=-1,0,1,2\):