6. If three angles $x, y, z$ form an arithmetic sequence, with a common difference of $\frac{\pi}{2}$, then $\tan x \tan y+\tan y \tan z+\tan z \tan x=$ $\qquad$
$$
\begin{array}{l}
\tan y(\tan x+\tan z)+\tan x \tan z \\
=\tan y\left(\frac{\tan y-\sqrt{3}}{1+\sqrt{3} \tan y}+\frac{\tan y+\sqrt{3}}{1-\sqrt{3} \tan y}\right)+\frac{\tan y-\sqrt{3}}{1+\sqrt{3} \tan y} \cdot \frac{\tan y+\sqrt{3}}{1-\sqrt{3} \tan y} \\
=\frac{8 \tan ^{2} y}{1-3 \tan ^{2} y}+\frac{\tan ^{2} y-3}{1-3 \tan ^{2} y}=-3 .
\end{array}
$$