If $\cot \left(-\frac{4\pi }{3}\right)=x$, then the value of $x$ is:

We need to evaluate $x=\cot \left(-\frac{4\pi }{3}\right)$.Using the property $\cot (-\theta )=-\cot (\theta )$:
$$\cot \left(-\frac{4\pi }{3}\right)=-\cot \left(\frac{4\pi }{3}\right)$$
Now, rewrite $\frac{4\pi }{3}$ as $\pi +\frac{\pi }{3}$:
$$\cot \left(\pi +\frac{\pi }{3}\right)=\cot \left(\frac{\pi }{3}\right)$$
(Since $\cot$ is positive in the third quadrant)Therefore:
$$x=-\cot \left(\frac{\pi }{3}\right)$$
We know that $\cot \left(\frac{\pi }{3}\right)=\frac{1}{\sqrt{3}}$. Thus:
$$x=-\frac{1}{\sqrt{3}}$$