Question

Prove $\frac{4\mathrm{sinAsinBsinC}}{4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}=8\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}.$

Answer 1

Consider the L.H.S.

$=\frac{4\sin A\sin B\sin C}{4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}$ ……. (1)

We know that

$\sin A=2\sin \frac{A}{2}\cos \frac{A}{2}$

From equation (1), we get

$=\frac{2\sin \frac{A}{2}cos\frac{A}{2}\times 2\sin \frac{B}{2}cos\frac{B}{2}\times 2\sin \frac{C}{2}cos\frac{C}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}$

$=8\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

R.H.S

Hence, proved.