Question:
Mark the correct alternative in each of the following:
In any $\triangle \mathrm{ABC}$, the value of $2 a c \sin \left(\frac{A-B+C}{2}\right)$ is
(a) $a^{2}+b^{2}-c^{2}$
(b) $c^{2}+a^{2}-b^{2}$
(c) $b^{2}-c^{2}-a^{2}$
(d) $c^{2}-a^{2}-b^{2}$
Solution:
In $\Delta \mathrm{ABC}$
$A+B+C=\pi \quad$ (Angle sum property)
$\Rightarrow A+C=\pi-B$
$\therefore 2 a c \sin \left(\frac{A-B+C}{2}\right)$
$=2 a c \sin \left(\frac{\pi-2 B}{2}\right)$
$=2 a c \sin \left(\frac{\pi}{2}-B\right)$
$=2 a c \cos B$
$=2 a c\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right) \quad$ (Using cosine rule)
$=c^{2}+a^{2}-b^{2}$
Hence, the correct answer is option (b).