Explanation

(b)
Given, $A + B = 180{^\circ} - C$
$\Rightarrow \ \frac{A}{2} + \frac{B}{2} = 90{^\circ} - \frac{C}{2}$
$\Rightarrow \ \tan\left( \frac{A}{2} + \frac{B}{2} \right) = \tan\left( 90{^\circ} - \frac{C}{2} \right)$
$\Rightarrow \ \ \frac{\tan{\frac{A}{2} + \tan\frac{B}{2}}}{1 - \tan{\frac{A}{2}\tan\frac{B}{2}}} = \cot\frac{C}{2}$
$\Rightarrow \ \ \left( \tan{\frac{A}{2} + \tan\frac{B}{2}} \right)\tan{\frac{C}{2} = 1 - \tan{\frac{A}{2}\tan\frac{B}{2}}}$
$\Rightarrow \ \tan{\frac{A}{2}\tan{\frac{B}{2} + \tan{\frac{B}{2}\tan{\frac{C}{2} + \tan{\frac{A}{2}\tan{\frac{C}{2} = 1}}}}}}$