The value of $$\frac{{2{{\cos }^3}\theta - \cos \theta }}{{\sin \theta - 2{{\sin }^3}\theta }}:$$

$$\eqalign{ & \frac{{2{{\cos }^3}\theta - \cos \theta }}{{\sin \theta - 2{{\sin }^3}\theta }} \cr & = \frac{{\cos \theta \left[ {2{{\cos }^2}\theta - 1} \right]}}{{\sin \theta \left[ {1 - 2{{\sin }^2}\theta } \right]}} \cr & = \frac{{\cos \theta \times \cos 2\theta }}{{\sin \theta \times \cos 2\theta }} \cr & = \cot \theta \cr} $$