(i) Start with the given equation:

\(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\)

Replace \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\):

\(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \cdot \frac{\sin \theta}{\cos \theta}\)

Multiply both sides by \(\cos \theta (\sin \theta + \cos \theta)\):

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

Expand both sides:

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

Subtract \(2 \sin \theta \cos \theta\) from both sides:

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

(ii) From \(\cos^2 \theta = 2 \sin^2 \theta\), use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\(\cos^2 \theta = 2 (1 - \cos^2 \theta)\)

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

\(3 \cos^2 \theta = 2\)

\(\cos^2 \theta = \frac{2}{3}\)

\(\cos \theta = \pm \sqrt{\frac{2}{3}}\)

For \(0^\circ < \theta < 180^\circ\), \(\theta = 35.3^\circ\) or \(144.7^\circ\).