Start by obtaining a common denominator for the two fractions:

\(\frac{(\sin \theta + 2 \cos \theta)(\cos \theta + 2 \sin \theta) - (\sin \theta - 2 \cos \theta)(\cos \theta - 2 \sin \theta)}{(\cos \theta - 2 \sin \theta)(\cos \theta + 2 \sin \theta)}\)

Expand the numerators:

\(5 \sin \theta \cos \theta + 2 \sin^2 \theta + 2 \cos^2 \theta - (5 \sin \theta \cos \theta - 2 \sin^2 \theta - 2 \cos^2 \theta)\)

Simplify the expression:

\(\frac{4(\cos^2 \theta + \sin^2 \theta)}{\cos^2 \theta - 4 \sin^2 \theta}\)

Use the identity \(\cos^2 \theta + \sin^2 \theta = 1\):

\(\frac{4}{\cos^2 \theta - 4 \sin^2 \theta}\)

Recognize that \(\cos^2 \theta - 4 \sin^2 \theta = 5 \cos^2 \theta - 4\) by using \(\cos^2 \theta + \sin^2 \theta = 1\) again:

\(\frac{4}{5 \cos^2 \theta - 4}\)