$\sin 3\theta . {\cos }^3\theta + \cos 3\theta . {\sin }^3\theta =$

$\begin{array}{rcl} & & \\ \cos 3\mathrm{\theta }& =& 4{\cos }^3\mathrm{\theta }-3\mathrm{cos\theta }\\ {\cos }^3\mathrm{\theta }& =& \frac{1}{4} (\cos 3\mathrm{\theta }+3\mathrm{cos\theta })\\ \sin 3\mathrm{\theta }& =& 3\mathrm{sin\theta }-4{\sin }^3\mathrm{\theta }\\ {\sin }^3\mathrm{\theta }& =& \frac{1}{4}(3\mathrm{sin\theta }-\sin 3\mathrm{\theta })\end{array}$

$\begin{array}{rcl} & & \sin 3\mathrm{\theta } {\cos }^3\mathrm{\theta }+\cos 3\mathrm{\theta } {\sin }^3\mathrm{\theta }\\ & =& \sin 3\mathrm{\theta }\frac{1}{4}(3\mathrm{cos\theta }+\cos 3\mathrm{\theta })+\frac{\cos 3\mathrm{\theta }}{4}(3\mathrm{sin\theta }-\sin 3\mathrm{\theta })\\ & =& \frac{3}{4}\left[\sin 3\mathrm{\theta } \mathrm{cos\theta }+\cos 3\mathrm{\theta } \mathrm{sin\theta }\right]+\frac{1}{4}\left(\sin 3\mathrm{\theta } \cos 3\mathrm{\theta }-\sin 3\mathrm{\theta } \cos 3\mathrm{\theta }\right)\\ & =& \frac{3}{4} \sin (3\mathrm{\theta }+\mathrm{\theta })\\ & =& \frac{3}{4} \sin 4\mathrm{\theta }\end{array}$