(i) Start with the equation:

\(\sqrt{2} \csc x + \cot x = \sqrt{3}\)

Express \(\csc x\) and \(\cot x\) in terms of sine and cosine:

\(\sqrt{2} \left( \frac{1}{\sin x} \right) + \frac{\cos x}{\sin x} = \sqrt{3}\)

Combine into a single fraction:

\(\frac{\sqrt{2} + \cos x}{\sin x} = \sqrt{3}\)

Multiply through by \(\sin x\):

\(\sqrt{2} + \cos x = \sqrt{3} \sin x\)

Rearrange to:

\(\sqrt{3} \sin x - \cos x = \sqrt{2}\)

Use the identity:

\(a \sin \theta - b \cos \theta \equiv R \sin(\theta - \alpha)\)

where \(R = \sqrt{a^2 + b^2}\) and \(\tan \alpha = \frac{b}{a}\).

Here, \(a = \sqrt{3}\) and \(b = 1\).

Calculate \(R\):

\(R = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2\)

Calculate \(\alpha\):

\(\tan \alpha = \frac{1}{\sqrt{3}}\)

\(\alpha = 30^\circ\)

Thus, \(\sqrt{3} \sin x - \cos x = 2 \sin(x - 30^\circ)\).

(ii) Solve \(2 \sin(x - 30^\circ) = \sqrt{2}\):

\(\sin(x - 30^\circ) = \frac{\sqrt{2}}{2}\)

\(x - 30^\circ = 45^\circ\) or \(x - 30^\circ = 135^\circ\)

\(x = 75^\circ\) or \(x = 165^\circ\)