In any $ riangle A B C, b^2 \sin 2 C+c^2 \sin B$ is equal to

Question

In any $	riangle A B C, b^2 \sin 2 C+c^2 \sin B$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & b^2 \sin 2 C+c^2 \sin 2 B \ & =2 b^2 \sin C \cos C+2 c^2 \sin B \cos B \ & =\frac{2 b^2 c \cos C}{2 R}+\frac{2 c^2 b \cos B}{2 R} \ & \qquad\left[\because \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 Right] \ & =\frac{b c}{R}(b \cos C+c \cos B)=\frac{a b c}{R} \ & {[\because 	ext { by projection formula } b \cos C+c \cos B=a]} \ & =4 \Delta \ & \quad\left[\because R=\frac{a b c}{4 \Delta}ight]\end{aligned}$

In any $\triangle A B C, b^2 \sin 2 C+c^2 \sin B$ is equal to

Solution