A true statement among the following identities is

We have,

$\sin \left(5\theta \right)=\sin \left(3\theta +2\theta \right)$

$\Rightarrow \sin \left(5\theta \right)=\sin 3\theta \cos 2\theta +\cos 3\theta \sin 2\theta$

$\Rightarrow \sin \left(5\theta \right)=\left(3\sin \theta -4{\sin }^3\theta \right)\left(2{\cos }^2\theta -1\right)+\left(4{\cos }^3\theta -3\cos \theta \right)\left(2\sin \theta \cos \theta \right)$

$\Rightarrow \sin \left(5\theta \right)=6\sin \theta {\cos }^2\theta -3\sin \theta -8{\sin }^3\theta {\cos }^2\theta +4{\sin }^3\theta +\left(8\sin \theta {\cos }^4\theta -6\sin \theta {\cos }^2\theta \right)$

$\Rightarrow \sin \left(5\theta \right)=-3\sin \theta -8{\sin }^2\theta \sin \theta {\cos }^2\theta +4{\sin }^2\theta \sin \theta +8\sin \theta {\cos }^4\theta$

$\Rightarrow \sin \left(5\theta \right)=-3\sin \theta -8\left(1-{\cos }^2\theta \right)\sin \theta {\cos }^2\theta +4\left(1-{\cos }^2\theta \right)\sin \theta +8\sin \theta {\cos }^4\theta$

$\Rightarrow \sin \left(5\theta \right)=-3\sin \theta -8\sin \theta {\cos }^2\theta +8\sin \theta {\cos }^4\theta +4\sin \theta -4\sin \theta {\cos }^2\theta +8\sin \theta {\cos }^4\theta$

$\Rightarrow \sin \left(5\theta \right)=16\sin \theta {\cos }^4\theta -12\sin \theta {\cos }^2\theta +\sin \theta$