Question

Express $\sin 12\theta +\sin 4\theta$ as the product of sines and cosines.

Answer 1

We have,

$\sin 12\theta +\sin 4\theta$

$\Rightarrow \sin 2\left(6\theta \right)+\sin 2\left(2\theta \right)$

$\Rightarrow 2\sin 6\theta \cos 6\theta +2\sin 2\theta \cos 2\theta$

$\Rightarrow 2\sin 2\left(3\theta \right)\cos 6\theta +4\sin \theta \cos \theta \cos 2\theta$

$\Rightarrow 4\sin 3\theta \cos 3\theta \cos 6\theta +4\sin \theta \cos \theta \cos 2\theta$

Hence, this is the answer.