The value of $$\frac{{\sec \theta \left( {\sin \theta - 2{{\sin }^3}\theta } \right)}}{{2{{\cos }^3}\theta - \cos \theta }}$$    is:

$$\eqalign{ & \frac{{\sec \theta \left( {\sin \theta - 2{{\sin }^3}\theta } \right)}}{{2{{\cos }^3}\theta - \cos \theta }} \cr & \frac{{\sec \theta .\sin \theta \left( {1 - 2{{\sin }^2}\theta } \right)}}{{\cos \theta \left( {2{{\cos }^2}\theta - 1} \right)}} \cr & \frac{{\sec \theta .\sin \theta \times \cos 2\theta }}{{\cos \theta \times \cos 2\theta }} \cr & \sec \theta .\tan \theta \cr} $$