If   $\alpha +\beta +\gamma =2\pi ,$  then 

$\alpha +\beta +\gamma =2\pi or\frac{\alpha }{2}+\frac{\beta }{2}+\frac{\gamma }{2}=\pi$

$\Rightarrow \therefore \tan \left(\frac{\alpha }{2}+\frac{\beta }{2}\right)=\tan \left(\pi -\frac{\gamma }{2}\right)=-\tan \frac{\gamma }{2}$

$\Rightarrow \frac{\tan {\displaystyle \frac{\alpha }{2}}+\tan {\displaystyle \frac{\beta }{2}}}{1-\tan \frac{\alpha }{2}\tan \frac{\beta }{2}}=-\tan \frac{\gamma }{2}$

$\Rightarrow \tan \frac{\alpha }{2}+\tan \frac{\beta }{2}+\tan \frac{\gamma }{2}=\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\tan \frac{\gamma }{2}$