10. In an oblique $\triangle A B C$, $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=\sin ^{2} B$, then $\tan A \tan C=$
$$
\begin{array}{l}
\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=\sin ^{2} B \Rightarrow \cos ^{2} A+\cos ^{2} C-1=-2 \cos ^{2} B \\
\Rightarrow \frac{1}{2}(\cos 2 A+\cos 2 C)=\cos (A+C) \cos (A-C)=-2 \cos ^{2}(A+C) \\
\Rightarrow \cos (A-C)=-2 \cos (A+C) \Rightarrow \sin A \sin C=3 \cos A \cos C \\
\Rightarrow \tan A \tan C=3 .
\end{array}
$$