geometry

Using the half-angle formulae:

                                                        $tan(A)=\frac{2tan(\frac{A}{2})}{1-tan^2(\frac{A}{2})}=\frac{2(\frac{1}{4})}{1-(\frac{1}{4})^2}=\frac{8}{15}$  .........(1)

                                                        $tan(B)=\frac{2tan(\frac{B}{2})}{1-tan^2(\frac{B}{2})}=\frac{2(\frac{1}{5})^2}{1-(\frac{1}{5})^2}=\frac{5}{12}$ ..........(2)

                                                                   $tan(C)=\frac{2tan(\frac{C}{2})}{1-tan^2(\frac{C}{2})}$                ........(*)

Since the angles in a triangle sum to 180 degrees, then:

                                              $A+B+C=180 \iff tan(A+B+C)=0$  (Take tan of both sides).

Thus, consider:                        $tan((A+B)+C)=\frac{tan(A+B)+tan(C)}{1-tan(A+B)tan(C)}=0$        ........(**)

Now consider $tan(A+B)$, and using (1) and (2) gives:

                                             $tan(A+B)=\frac{tan(A)+tan(B}{1-tan(A)tan(B)}=\frac{\frac{8}{15}+\frac{5}{12}}{1-(\frac{8}{15}*\frac{5}{12})}=\frac{171}{140}$      ......(***)

Thus, substituting (***) in (**) gives:

                                                $\frac{171}{140}+tan(C)=0 \iff tan(C)=-\frac{171}{140}$

Next, consider (*) and let  $t=tan(\frac{C}{2})$

Then, 

                                            $-\frac{171}{140}=\frac{2t}{1-t^2} \iff 171t^2-171=280t$

Then,                              $171t^2-280t-171=0 \iff(19t+9)(9t-19)=0$

Thus,                                                       $t_1=\frac{19}{9},t_2=-\frac{9}{19}$

But, C is a positive angle (Angle in a triangle), hence $t=tan(\frac{C}{2})>0$

Therefore, only $t_1=\frac{19}{9}$ is the desired solution.

Thus,   $tan(\frac{C}{2})=\frac{19}{9}$

Hope this helps, and I think there is a simpler solution than this!