Simplify the expression:
$\dfrac{\cos 2 \alpha}{\sin^2 2 \alpha \left(\cos^2 \alpha - \tan^2 \alpha\right)}$
$\dfrac{\cos 2 \alpha}{\sin^2 2 \alpha \left(\cos^2 \alpha - \tan^2 \alpha\right)}$
$= \dfrac{\cos^2 \alpha - \sin^2 \alpha}{\left(2 \sin \alpha \cos \alpha\right)^2 \times \left(\dfrac{\cos^2 \alpha}{\sin^2 \alpha} - \dfrac{\sin^2 \alpha}{\cos^2 \alpha}\right)}$
$= \dfrac{\left(\cos^2 \alpha - \sin^2 \alpha\right) \times \sin^2 \alpha \cos^2 \alpha}{4 \sin^2 \alpha \cos^2 \alpha \times \left(\cos^4 \alpha - \sin^4 \alpha\right)}$
$= \dfrac{\cos^2 \alpha - \sin^2 \alpha}{4 \times \left[\left(\cos^2 \alpha\right)^2 - \left(\sin^2 \alpha\right)^2\right]}$
$= \dfrac{\cos^2 \alpha - \sin^2 \alpha}{4 \times \left(\cos^2 \alpha + \sin^2 \alpha\right) \times \left(\cos^2 \alpha - \sin^2 \alpha\right)}$
$= \dfrac{1}{4 \times 1} = \dfrac{1}{4}$