Question

Prove
SinA sin(pi/3-A) sin(pi/3+A) = 1/4 sin3A

Answer 1

Step-by-step explanation:

We have,

$\sin(A) \sin \bigg( \frac{\pi}{3} - A\bigg) \sin \bigg( \frac{\pi}{3} + A\bigg)$

$= \sin(A) \bigg[ \bigg \{ \sin \bigg( \frac{\pi}{3} \bigg) \cos(A) - \cos \bigg( \frac{\pi}{3} \bigg) \sin(A)\bigg \} \bigg \{ \sin \bigg( \frac{\pi}{3} \bigg) \cos(A) + \cos \bigg( \frac{\pi}{3} \bigg) \sin(A) \bigg\}\bigg]$

$= \sin(A) \bigg[ \sin ^{2} \bigg( \frac{\pi}{3} \bigg) \cos^{2} (A) - \cos^{2} \bigg( \frac{\pi}{3} \bigg) \sin ^{2} (A) \bigg]$

$= \sin(A) \bigg[ \frac{3}{4} \cos^{2} (A) - \frac{1}{4} \sin ^{2} (A) \bigg]$

$= \frac{1}{4} \sin(A) \bigg[ 3 \cos^{2} (A) - \sin ^{2} (A) \bigg]$

$= \frac{1}{4} \sin(A) \bigg[ 3 - 3\sin^{2} (A) - \sin ^{2} (A) \bigg]$

$= \frac{1}{4} \sin(A) (3 - 4\sin ^{2} (A))$

$= \frac{1}{4} (3\sin(A) - 4\sin ^{3} (A))$

$= \frac{1}{4}\sin(3A)$