Question

In a $\Delta ABC(\frac{a^2}{\sin A}+\frac{b^2}{\sin B}+\frac{c^2}{\sin C})\cdot \sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$ simplifies to

• A. $2\Delta$
• B. $\Delta$
• C. $\frac{\Delta }{2}$
• D. $\frac{\Delta }{4}$

Answer 1

Answer: (B)

$\Delta$

$(\frac{a^2}{\sin A}+\frac{b^2}{\sin B}+\frac{c^2}{\sin C})\cdot \sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$
$(\frac{4R^2{\sin }^{2}A}{\sin A}+\frac{4R^2{\sin }^{2}B}{\sin B}+\frac{4R^2{\sin }^{2}C}{\sin C})\prod \sin \frac{A}{2}$ ($\because \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$)
$=4R^2(\sin A+\sin B+\sin C)\prod \sin \frac{A}{2}$
$=16R^2\prod \cos \frac{A}{2}.\prod \sin \frac{A}{2}$ ($\because \sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{A}{2}$ )
$=2R^2\sin A.\sin B.\sin C$
$=2R^2\frac{abc}{8R^3}$
$=\frac{abc}{4R}=\Delta$