$\frac{{\cos }^3\theta -\cos 3\theta }{\cos \theta }+\frac{{\sin }^3\theta +\sin 3\theta }{\sin \theta }=$

$\frac{{\cos }^3\theta -\cos 3\theta }{\cos \theta }+\frac{{\sin }^3\theta +\sin 3\theta }{\sin \theta }$

$\frac{{\cos }^3\theta -3{\cos }^3\theta +3\cos \theta }{\cos \theta }+\frac{{\sin }^3\theta +3\sin \theta -4{\sin }^3\theta }{\sin \theta }$

$\frac{3\cos \theta -3{\cos }^3\theta }{\cos \theta }+\frac{3\sin \theta -3{\sin }^3\theta }{\sin \theta }$

$=3-3{\cos }^2\theta +3-3{\sin }^2\theta$

$=6-3({\cos }^2\theta +{\sin }^2\theta )$

$=6-3\left(1\right)$

$=3$