Question

In $\Delta ABC,\sum \frac{a^{2}sin(B-C)}{sinA}=$

• A. $a^2{b}^2$
• B. $b^2{c}^2$
• C. $c^2{a}^2$
• D. $0$

Answer 1

The correct option is D $0$

In $△ABC,a=2R\sin A=2R\sin ({180}^{\circ }-(B+C\left)\right)=2R\sin (B+C),\frac{a}{\sin A}=2R$

$\sum \frac{a^2\sin (B-C)}{\sin A}=\sum \frac{a}{\sin A}\times a\times \sin (B-C)=\sum \frac{a}{\sin A}\left(2R\sin \right(B+C\left)\sin \right(B-C\left)\right)$

$=4R^2\sum ({\sin }^{2}B-{\sin }^{2}C)$ $(\because \sin (A+B\left)\sin \right(A-B)={\sin }^{2}A-{\sin }^{2}B)$

$=4R^2({\sin }^{2}B-{\sin }^{2}C+{\sin }^{2}C-{\sin }^{2}A+{\sin }^{2}A-{\sin }^{2}B)$

$=0$