8.2. For all triples $(x, y, z)$ satisfying the system

$$
\left\{\begin{array}{l}
\sqrt{3} \sin x=\operatorname{tg} y \\
2 \sin y=\operatorname{ctg} z \\
\sin z=2 \operatorname{tg} x
\end{array}\right.
$$

find the smallest value of the expression $\cos x-\cos z$.
Answer. $-\frac{7 \sqrt{2}}{6} \approx-1.65$.