5. In $\triangle A B C$, if $\tan A \tan B=\tan A \tan C+\tan C \tan B$, then $\frac{a^{2}+b^{2}}{c^{2}}=$
5. 3 .

Given the equation
$$
\frac{\sin A \cdot \sin B}{\cos A \cdot \cos B}=\frac{\sin A \cdot \sin C}{\cos A \cdot \cos C}+\frac{\sin B \cdot \sin C}{\cos B \cdot \cos C},
$$

Multiplying both sides by $\cos A \cdot \cos B \cdot \cos C$, we get
$$
\sin A \cdot \sin B \cdot \sin C=\sin C(\sin A \cdot \cos B+\cos A \cdot \sin B)=\sin ^{2} C \text {. }
$$

Therefore,
$$
a b \cdot \frac{a^{2}+b^{2}-c^{2}}{2 a b}=c^{2} .
$$

Thus,
$$
a^{2}+b^{2}=3 c^{2} \Rightarrow \frac{a^{2}+b^{2}}{c^{2}}=3 .
$$