Trigonometric Functions — Maths STD 12 Science — Question

$\begin{aligned} & \frac{a}{\sin \mathrm{A}}=\frac{b}{\sin \mathrm{B}}=\frac{c}{\sin C}=k \\ & \therefore a=k \sin A, b=k \sin B, c=k \sin C\end{aligned}$

$\begin{aligned} & \text { RHS }=\left(b^2-c^2\right) \sin \mathrm{A} \\ & =\left(k^2 \sin ^2 \mathrm{~B}-k^2 \sin ^2 C\right) \sin \mathrm{A}\end{aligned}$

$\begin{aligned} & =k^2\left(\sin ^2 B-\sin ^2 C\right) \sin A \\ & =k^2(\sin B+\sin C)(\sin B-\sin C) \sin A\end{aligned}$

$\begin{aligned} & =k^2 \times 2 \sin \left(\frac{B+C}{2}\right) \cdot \cos \left(\frac{B-C}{2}\right) \times \\ & 2 \cos \left(\frac{B+C}{2}\right) \cdot \sin \left(\frac{B-C}{2}\right) \times \sin A\end{aligned}$

$\begin{aligned} & =k^2 \times 2 \sin \left(\frac{B+C}{2}\right) \cdot \cos \left(\frac{B+C}{2}\right) \times \\ & 2 \sin \left(\frac{B-C}{2}\right) \cdot \cos \left(\frac{B-C}{2}\right) \times \sin A\end{aligned}$

$\begin{aligned} & =k^2 \times \sin (B+C) \times \sin (B-C) \times \sin A \\ & =k^2 \cdot \sin (\pi-A) \cdot \sin (B-C) \cdot \sin A \ldots[\because A+B+C=\pi]\end{aligned}$

$\begin{aligned} & =k^2 \sin A \cdot \sin (B-C) \cdot \sin A \\ & =(k \sin A)^2 \cdot \sin (B-C) \\ & =a^2 \sin (B-C)=L H S .\end{aligned}$