Question:
The value of $108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is
Solution:
$108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$
$=36\left(3 \sin \frac{\pi}{9}-4 \sin ^{3} \frac{\pi}{9}\right)$
$=36\left(\sin 3\left(\frac{\pi}{9}\right)\right)$ $\left[\right.$ using identity $\left.\because 3 \sin x-4 \sin ^{3} x=\sin 3 x\right]$
$=36\left(\sin \frac{\pi}{3}\right)$
$=36\left(\frac{\sqrt{3}}{2}\right)$
$=18 \sqrt{3}$
$\therefore 108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is $18 \sqrt{3}$