Answer
$sin3\theta+sin\theta=2sin2\theta$ $cos\theta$
Work Step by Step
Need to prove the identity
$sin3\theta+sin\theta=2sin2\theta$ $cos\theta$
Since $sin3\theta$ can be written as:
$sin3\theta=sin(2\theta+\theta)$
Use sum identity for sine.
$sin3\theta=sin(2\theta+\theta)=sin2\theta cos\theta+cos2\theta sin\theta$
Thus,
$sin3\theta+sin\theta=sin2\theta cos\theta+cos2\theta sin\theta+sin\theta$
$=sin2\theta cos\theta+(2cos^{2}\theta-1) sin\theta+sin\theta$
$=sin2\theta cos\theta+2cos^{2}\theta sin\theta$
$=sin2\theta cos\theta+(2 sin\theta cos\theta) cos \theta$
$ =sin2\theta cos\theta+ sin2\theta cos \theta $
Hence, $sin3\theta+sin\theta=2sin2\theta$ $cos\theta$
