Question

State whether the following statements are true or false:
$a\cos A+b\cos B+c\cos C=4R\sin A\sin B\sin C=(a+b+c)\frac{r}{R}$

• A. True
• B. False

Answer 1

The correct option is A True

$a=2R\sin A$
LHS $=R(\sin 2A+\sin 2B+\sin 2C)=4R\sin 2A+\sin 2B+\sin 2C=$ middle term
Again $(a+b+c)\frac{r}{R}=2R(\sin A+\sin B+\sin C)\times \frac{4R\sin \left(A/2\right)\sin \left(B/2\right)\sin \left(C/2\right)}{R}$
$=4R\sin A\sin B\sin C$
$\because \sin A=2\sin \frac{A}{2}\cos \frac{A}{2}$