If \(A+B+C=180^{\circ}\), then the value of
\(\tan \left(\frac{\mathrm{A}}{2}\right) \tan \left(\frac{\mathrm{B}}{2}\right)+\tan \left(\frac{\mathrm{B}}{2}\right) \tan \left(\frac{\mathrm{C}}{2}\right)+\tan \left(\frac{\mathrm{C}}{2}\right) \tan \left(\frac{\mathrm{A}}{2}\right)\) is

\(\ln \Delta A B C, A+B+C=\pi \Rightarrow A+B=\pi-C\)
\(\tan \left(\frac{A+B}{2}\right)=\tan \left(\frac{\pi-C}{2}\right)=\tan \left(\frac{\pi}{2}-\frac{C}{2}\right)\)
\(\therefore \frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2} \tan \frac{B}{2}}=\cot \frac{C}{2} \Rightarrow \frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2} \cdot \tan \frac{B}{2}}=\frac{1}{\tan \frac{C}{2}}\)
\(\therefore \tan \frac{\mathrm{C}}{2}\left(\tan \frac{\mathrm{A}}{2}+\tan \frac{\mathrm{B}}{2}\right) \quad=1-\tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}\)
\(\therefore \tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}+\tan \frac{\mathrm{B}}{2} \tan \frac{\mathrm{C}}{2}+\tan \frac{\mathrm{C}}{2} \tan \frac{\mathrm{A}}{2}=1\)