LHS
= \sin 3 \theta = \sin ( \theta + 2\theta) = \ sin \theta \cos 2 \theta + \cos \theta \sin 2\theta usng \sin(A+B) formula
= \ sin \theta \cos 2 \theta + \cos \theta (2\sin \theta \cos \theta) using \sin 2\theta formula
= \ sin \theta (\cos 2 \theta + 2 \cos^2 \theta)
= \ sin \theta ( 2 \cos 2 \theta + \cos 2 \theta + 1 ) using formula for \cos 2\theta
= \ sin \theta ( 2 \cos 2 \theta + 1 )
= 2 \ sin \theta \cos 2 \theta + \sin \theta which is RHS
In the above putting \theta = \frac{\pi}{3} we get
\sin \frac{\pi}{9} + 2 \sin \frac{\pi}{9} \cos \frac{2\pi}{9} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}
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