In $\Delta ABC$if a, b, c are in A.P., then $\tan \frac{A}{2}\tan \frac{C}{2}=$

$\tan \frac{A}{2}\tan \frac{C}{2}=\frac{\Delta }{s(s-a)}.\frac{\Delta }{s(s-c)}=\frac{{\Delta }^2}{s.s(s-a)(s-c)}$

$=\frac{{\Delta }^2(s-b)}{s.s(s-a)(s-b)(s-c)}$

$=\frac{s-b}{s}$

a ,b , c  are A.P

$\begin{array}{l}2b=a+c\\ \frac{a+c-b}{a+b+c}=\frac{2b-b}{2b+b}=\frac{b}{3b}=\frac{1}{3}\end{array}$