Question

In $△ABC$, if $a,b,c$ are in A.P., then the value of $\frac{\sin \frac{A}{2}\sin \frac{C}{2}}{\sin \frac{B}{2}}=$

Answer 1

$\frac{\sin \frac{A}{2}\sin \frac{C}{2}}{\sin \frac{B}{2}}$

$=\frac{\sqrt{\frac{(s-b)(s-c)}{bc}}\times \sqrt{\frac{(s-a)(s-b)}{ab}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}}$ ( where $s=\frac{a+b+c}{2}$ )

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

$=\frac{s}{b}-1$

$=\frac{a+b+c}{2b}-1$

$=\frac{2b+b}{2b}-1$ ( a,b,c are in A.P. so $a+c=2b$ )

$=\frac{3}{2}-1=\frac{1}{2}$