(i) Start with the equation:

\(\frac{\cos \theta - 4}{\sin \theta} - \frac{4 \sin \theta}{5 \cos \theta - 2} = 0\)

Multiply through by \(\sin \theta (5 \cos \theta - 2)\) to clear the denominators:

\((\cos \theta - 4)(5 \cos \theta - 2) - 4 \sin^2 \theta = 0\)

Expand and simplify:

\(5 \cos^2 \theta - 22 \cos \theta + 8 - 4(1 - \cos^2 \theta) = 0\)

\(5 \cos^2 \theta - 22 \cos \theta + 8 - 4 + 4 \cos^2 \theta = 0\)

Combine like terms:

\(9 \cos^2 \theta - 22 \cos \theta + 4 = 0\)

Thus, the equation is shown as required.

(ii) Solve the quadratic equation:

\(9 \cos^2 \theta - 22 \cos \theta + 4 = 0\)

Using the quadratic formula \(\cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -22\), \(c = 4\):

\(\cos \theta = \frac{22 \pm \sqrt{(-22)^2 - 4 \times 9 \times 4}}{18}\)

\(\cos \theta = \frac{22 \pm \sqrt{484 - 144}}{18}\)

\(\cos \theta = \frac{22 \pm \sqrt{340}}{18}\)

\(\cos \theta = \frac{22 \pm 18.44}{18}\)

\(\cos \theta = 0.1978 \text{ or } 2.247\)

Since \(\cos \theta\) must be between -1 and 1, only \(\cos \theta = 0.1978\) is valid.

Find \(\theta\):

\(\theta = \cos^{-1}(0.1978) \approx 78.6^\circ\)

For the second solution in the range \(0^\circ \leq \theta \leq 360^\circ\), use \(360^\circ - 78.6^\circ = 281.4^\circ\).

Thus, \(\theta = 78.6^\circ, 281.4^\circ\).