If ${\sin }^2\frac{A}{2}{\sin }^2\frac{B}{2}{\sin }^2\frac{C}{2}$  are in H.P., then show that $a,b,c$are in H.P. 

 ${\sin }^2\frac{A}{2},{\sin }^2\frac{B}{2},{\sin }^2\frac{C}{2}$ are in H.P. 

$\Rightarrow \frac{bc}{(s-b)(s-c)},\frac{ca}{(s-c)(s-a)},\frac{ab}{(s-a)(s-b)}$  are  in H.P. 

$\Rightarrow \frac{2ca}{(s-c)(s-a)}=\frac{bc}{(s-b)(s-c)}+\frac{ab}{(s-a)(s-b)}$

$\Rightarrow \frac{2ca}{(s-c)(s-a)}=\frac{bc(s-a)+ab(s-c)}{(s-a)(s-b)(s-c)}$

$\Rightarrow 2ca(s-b)=bc(s-a)+ab(s-c)$

$\Rightarrow 2cas-2abc=bcs-abc+abs-abc$

$\Rightarrow 2cas=sb(c+a)$

$b=\frac{2ac}{a+c}$     $\therefore a,b,c$  are in H.P.