In a triangle ABC , with usual notations, $2 \mathrm{ac} \sin…

In a triangle ABC , with usual notations, $2 \mathrm{ac} \sin \left(\frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2}\right)$ is equal to
  1. $a^2+b^2-c^2$
  2. $b^2-a^2+c^2$
  3. $\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2$
  4. $\mathrm{a}^2-\mathrm{b}^2-\mathrm{c}^2$

Solution

$\begin{aligned} 2 \mathrm{ac} \sin \frac{\mathrm{A}-\mathrm{B}+\mathrm{C}}{2} & =2 \mathrm{ac} \sin \frac{\pi-2 \mathrm{~B}}{2} \\ & =2 \mathrm{ac} \cos \mathrm{B} \\ & =2 \mathrm{ac} \frac{\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2}{2 \mathrm{ca}} \\ & =\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\end{aligned}$....[By cosine rule]

Asked in: MHT CET 2024 (16 May Shift 1)

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