(i) Start with the expression:

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

Combine the fractions:

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

Expand the numerator:

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

Which simplifies to:

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

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

\(\frac{1}{\sin^2 \theta - \cos^2 \theta}\)

(ii) Solve the equation:

\(\frac{1}{\sin^2 \theta - \cos^2 \theta} = 3\)

\(\sin^2 \theta - \cos^2 \theta = \frac{1}{3}\)

Using \(\sin^2 \theta + \cos^2 \theta = 1\), let \(s = \sin \theta\) and \(c = \cos \theta\):

\(s^2 - c^2 = \frac{1}{3}\)

\(s^2 + c^2 = 1\)

Subtract the equations:

\(2c^2 = \frac{2}{3}\)

\(c^2 = \frac{1}{3}\)

\(c = \pm \frac{1}{\sqrt{3}}\)

\(s^2 = \frac{2}{3}\)

\(s = \pm \sqrt{\frac{2}{3}}\)

\(\tan \theta = \pm \sqrt{2}\)

Solutions for \(\theta\) are:

\(\theta = 54.7^\circ, 125.3^\circ, 234.7^\circ, 305.3^\circ\)