Question

In any $\Delta ABC,$ $2[a{\sin }^2\left(\frac{C}{2}\right)+c{\sin }^2\left(\frac{A}{2}\right)]$ equals

• A. $a+c-b$
• B. $a-c+b$
• C. $a+c+b$
• D. $a-c-b$

Answer 1

The correct option is A $a+c-b$

$2[a{\sin }^2\left(\frac{C}{2}\right)+c{\sin }^2\left(\frac{A}{2}\right)]$
$=\left[a\right(1-\cos C)+c(1-\cos A\left)\right]$
$=a+c-(a\cos C+c\cos A)$
$=a+c-b$$\text{[By projection formulae]}$