Question

Express $a{\sin }^2\frac{C}{2}+c{\sin }^2\frac{A}{2}$ interms of $s,a,b,c$.

Answer 1

So, $a\left[\frac{1-\cos C}{2}\right]+C\left(\frac{1-\cos A}{2}\right)$

$\Rightarrow \frac{a}{2}-\frac{a}{2}\cos C+\frac{C}{2}-\frac{C}{2}\cos A$

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

$=(\left(\frac{a+c}{2}\right)-\frac{b}{2})$

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

$\cos 2x=1-2{\sin }^{2}x$

$\Rightarrow \cos x=1-2{\sin }^2\frac{x}{2}$

${\sin }^2\frac{x}{2}=\frac{1-\cos x}{2}$

$1-\cos 2\theta =2{\sin }^2\theta$

Replace $\left(\theta by\frac{\theta }{2}\right)$

$1-\cos \theta =2{\sin }^2\frac{\theta }{2}$

${\sin }^2\frac{\theta }{2}=\frac{1-\cos \theta }{2}$.