(a) To show the given identity, start by obtaining a common denominator:

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

Expanding the numerators:

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

\(= 4(\cos^2 \theta + \sin^2 \theta)\)

Using \(\cos^2 \theta + \sin^2 \theta = 1\), the expression simplifies to:

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

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

(b) For the equation:

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

Multiply both sides by \(5 \cos^2 \theta - 4\):

\(4 = 5(5 \cos^2 \theta - 4)\)

\(25 \cos^2 \theta = 24\)

\(\cos^2 \theta = \frac{24}{25}\)

\(\cos \theta = \pm \sqrt{\frac{24}{25}} = \pm 0.9798\)

For \(0^\circ < \theta < 180^\circ\), \(\theta = 11.5^\circ\) or \(168.5^\circ\).