Answer
$\sin 3 \theta =\sin \theta (4 \cos^2 \theta-1)$
Work Step by Step
Re-arrange as: $\sin ( 2 \theta +\theta) = \sin 2 \theta \cos \theta +\cos 2 \theta \sin \theta$
or, $=2 \sin \theta \cos \theta \times \cos \theta +(1-2 \sin^2 \theta) \times \sin \theta$
or, $=\sin \theta ( 2 \cos^2 \theta+1-2 \sin^2 \theta)$
or, $=\sin \theta [ 2 \cos^2 \theta+1-2 (1-\cos^2 \theta)]$
Thus, it has been proven that $\sin 3 \theta =\sin \theta (4 \cos^2 \theta-1)$
