Mark the correct alternative in each of the following:
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).