Question

Express $2asi{n}^2\frac{C}{2}+csi{n}^2\frac{A}{2}$ in terms of s, a, b, c.

Answer 1

$asi{n}^2\frac{C}{2}+csi{n}^2\frac{A}{2}$

$=a\left(\frac{1-\cos C}{2}\right)+c\left(\frac{1-\cos A}{2}\right)$

From prejection formula,

$b=a\cos C+c\cos A$

$=\frac{1}{2}[a-a\cos C+c-c\cos A]$

$=\frac{1}{2}[a+c-(a\cos C+c\cos A]$

$=\frac{1}{2}[a+c-b]$ [because Let $a+b+c=2s]$

$=\frac{1}{2}[2s-b-b]$

$=s-b$