$$\left(\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta}\right)^2 + 1, \theta$$ ≠ $$45^\circ$$ is equal to:

As per the given question,

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

Now, $$\Rightarrow \dfrac{\sin^2\theta}{\cos^2\theta} \left(\frac{1- 2 \sin^2 \theta}{2 \cos^2 \theta - 1}\right)^2 + 1$$

We know that $$\Rightarrow \cos 2\theta=2\cos^2\theta =1-2\sin^2\theta=\cos^2\theta-\sin^2\theta$$

Now, $$\Rightarrow \dfrac{\sin^2\theta}{\cos^2\theta} \left(\frac{\cos 2 \theta}{\cos2\theta}\right)^2 + 1$$

$$\Rightarrow \dfrac{\sin^2\theta}{\cos^2\theta}  + 1$$

Now taking LCM ,

$$\Rightarrow \dfrac{\sin^2\theta+\cos^2\theta}{\cos^2\theta}=\dfrac{1}{\cos^2\theta}=\sec^2\theta$$