5. In an oblique triangle $ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If $\frac{\tan C}{\tan A} + \frac{\tan C}{\tan B} = 1$, then $\frac{a^{2} + b^{2}}{c^{2}} =$
5. 3

Analysis: From $\frac{\tan C}{\tan A}+\frac{\tan C}{\tan B}=1$, we know that $\frac{\sin ^{2} C}{\cos C}=\sin A \sin B$. By the Law of Sines, $c^{2}=a b \cos C$, then $\frac{a^{2}+b^{2}}{c^{2}}=\frac{a^{2}+b^{2}-c^{2}}{c^{2}}+1=\frac{2 a b \cos C}{a b \cos C}+1=3$