Question

prove

$\tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}=1$

Answer 1

$A+B+C=\pi \Rightarrow \frac{A+B+C}{2}=\frac{\pi }{2}\Rightarrow \frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2}$

$\tan \left(\frac{A}{2}+\frac{B}{2}\right)=\cot \frac{C}{2}=\frac{1}{\tan \frac{C}{2}}$

$\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}.\tan \frac{B}{2}}=\frac{1}{\tan \frac{C}{2}}$

$\tan \frac{A}{2}.\tan \frac{C}{2}+\tan \frac{B}{2}.\tan \frac{C}{2}=1-\tan \frac{A}{2}.\tan \frac{B}{2}$

$\tan \frac{A}{2}.\tan \frac{C}{2}+\tan \frac{B}{2}.\tan \frac{C}{2}+\tan \frac{A}{2}.\tan \frac{B}{2}=1$