(i) Expand the left-hand side: \((\sin \theta + 2 \cos \theta)(1 + \sin \theta - \cos \theta) = \sin \theta + \sin^2 \theta - \sin \theta \cos \theta + 2 \cos \theta + 2 \cos \theta \sin \theta - 2 \cos^2 \theta\).

Rearrange terms: \(\sin \theta + \sin^2 \theta - \sin \theta \cos \theta + 2 \cos \theta + 2 \cos \theta \sin \theta - 2 \cos^2 \theta = \sin \theta + 2 \cos \theta + \sin^2 \theta + \sin \theta \cos \theta - 2 \cos^2 \theta\).

Use \(\sin^2 \theta = 1 - \cos^2 \theta\): \(\sin \theta + 2 \cos \theta + (1 - \cos^2 \theta) + \sin \theta \cos \theta - 2 \cos^2 \theta = \sin \theta + 2 \cos \theta + 1 - \cos^2 \theta + \sin \theta \cos \theta - 2 \cos^2 \theta\).

Simplify: \(1 + \sin \theta + 2 \cos \theta - 3 \cos^2 \theta + \sin \theta \cos \theta = 0\).

Rearrange: \(3 \cos^2 \theta - 2 \cos \theta - 1 = 0\).

(ii) Solve \(3 \cos^2 \theta - 2 \cos \theta - 1 = 0\).

Let \(x = \cos \theta\), then \(3x^2 - 2x - 1 = 0\).

Factorize: \((3x + 1)(x - 1) = 0\).

Solutions: \(x = 1\) or \(x = -\frac{1}{3}\).

For \(\cos \theta = 1\), \(\theta = 0^\circ\).

For \(\cos \theta = -\frac{1}{3}\), \(\theta = 109.5^\circ\) or \(\theta = -109.5^\circ\).