(i) Start with \(\left( \frac{1}{\sin \theta} - \frac{1}{\tan \theta} \right)^2\).

Rewrite \(\frac{1}{\tan \theta}\) as \(\frac{\cos \theta}{\sin \theta}\), so:

\(\left( \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} \right)^2 = \left( \frac{1 - \cos \theta}{\sin \theta} \right)^2\).

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

Using \(\sin^2 \theta = 1 - \cos^2 \theta\), we have:

\(\frac{(1 - \cos \theta)(1 - \cos \theta)}{1 - \cos^2 \theta} = \frac{1 - \cos \theta}{1 + \cos \theta}\).

(ii) Use the result from part (i):

\(\frac{1 - \cos \theta}{1 + \cos \theta} = \frac{2}{5}\).

Cross-multiply to get:

\(5(1 - \cos \theta) = 2(1 + \cos \theta)\).

Simplify to find \(\cos \theta\):

\(5 - 5\cos \theta = 2 + 2\cos \theta\).

\(3 = 7\cos \theta\).

\(\cos \theta = \frac{3}{7}\).

Find \(\theta\) using inverse cosine:

\(\theta = 64.6^\circ\) or \(295.4^\circ\).