4. If three angles $x, y, z$ form an arithmetic sequence with a common difference of $\frac{\pi}{3}$, then $\tan x \tan y+\tan y \tan z+\tan z \tan x=$ $\qquad$
Given $x=y-\frac{\pi}{3}, z=y+\frac{\pi}{3}$, then $\tan z=\frac{\tan y+\sqrt{3}}{1-\sqrt{3} \tan y}, \tan x=\frac{\tan y-\sqrt{3}}{1+\sqrt{3} \tan y}$, so
$$
\tan x \tan y=\frac{\tan ^{2} y-\sqrt{3} \tan y}{1+\sqrt{3} \tan y}, \tan y \tan z=\frac{\tan ^{2} y+\sqrt{3} \tan y}{1-\sqrt{3} \tan y}, \tan z \tan x=\frac{\tan ^{2} y-3}{1-3 \tan ^{2} y}
$$

Then $\tan x \tan y+\tan y \tan z+\tan z \tan x=\frac{9 \tan ^{2} y-3}{1-3 \tan ^{2} y}=-3$.