(i) Start with the given equation:

\(\sin(\theta + 45^\circ) + 2 \cos(\theta + 60^\circ) = 3 \cos \theta\)

Use angle addition formulas:

\(\sin(\theta + 45^\circ) = \sin \theta \cos 45^\circ + \cos \theta \sin 45^\circ\)

\(\cos(\theta + 60^\circ) = \cos \theta \cos 60^\circ - \sin \theta \sin 60^\circ\)

Substitute \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):

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

Simplify:

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

Combine terms:

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

Rearrange:

\(\left( \frac{\sqrt{2}}{2} - \sqrt{3} \right) \sin \theta = \left( 3 - \frac{\sqrt{2}}{2} - 1 \right) \cos \theta\)

\(\left( \frac{\sqrt{2}}{2} - \sqrt{3} \right) \sin \theta = \left( 2 - \frac{\sqrt{2}}{2} \right) \cos \theta\)

Divide both sides by \(\cos \theta\):

\(\tan \theta = \frac{2\sqrt{2} - 1}{1 - \sqrt{6}}\)

(ii) Solve for \(\theta\):

Using the expression for \(\tan \theta\), find \(\theta\) in the range \(0^\circ < \theta < 360^\circ\).

\(\theta = 128.4^\circ\) and \(\theta = 308.4^\circ\)

To solve the equation \(\sin(\theta - 30^\circ) + \cos \theta = 2 \sin \theta\), we start by using the angle subtraction identity:

\(\sin(\theta - 30^\circ) = \sin \theta \cos 30^\circ - \cos \theta \sin 30^\circ\)

Substitute the values \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\):

\(\sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2} + \cos \theta = 2 \sin \theta\)

Simplify the equation:

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

Combine like terms:

\(\frac{\sqrt{3}}{2} \sin \theta + \frac{1}{2} \cos \theta = 2 \sin \theta\)

Rearrange to form:

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

Factor out \(\sin \theta\):

\(\sin \theta \left( \frac{\sqrt{3}}{2} - 2 \right) = -\frac{1}{2} \cos \theta\)

Divide both sides by \(\cos \theta\):

\(\tan \theta = \frac{1}{4 - \sqrt{3}}\)

Calculate \(\theta\):

\(\theta = \tan^{-1}\left(\frac{1}{4 - \sqrt{3}}\right) \approx 23.8^\circ\)

Thus, the solution is \(\theta = 23.8^\circ\).

We solve the equation \(\cot \theta + \cot(\theta + 45^\circ) = 2\) using the addition formula for tangent.

Let \(t = \tan \theta\). Using the identity

\[ \tan(\theta + 45^\circ) = \frac{\tan \theta + \tan 45^\circ}{1 - \tan \theta \tan 45^\circ} = \frac{t + 1}{1 - t}, \] so \[ \cot(\theta + 45^\circ) = \frac{1}{\tan(\theta + 45^\circ)} = \frac{1 - t}{t + 1}. \]

Also, \(\cot \theta = \frac{1}{t}\). Substituting these into the original equation gives

\[ \frac{1}{t} + \frac{1 - t}{t + 1} = 2. \]

Multiply through by \(t(t+1)\):

\[ (t+1) + t(1 - t) = 2t(t+1). \]

Expanding and simplifying:

\[ t + 1 + t - t^2 = 2t^2 + 2t \quad \Rightarrow \quad 2t - t^2 + 1 = 2t^2 + 2t. \]

Bring all terms to one side:

\[ 0 = 3t^2 - 1 \quad \Rightarrow \quad t^2 = \frac{1}{3}. \]

So

\[ t = \pm \frac{1}{\sqrt{3}}. \]

This gives \(\tan \theta = \frac{1}{\sqrt{3}}\) or \(\tan \theta = -\frac{1}{\sqrt{3}}\).

In degrees:

• \(\tan \theta = \frac{1}{\sqrt{3}}\) gives \(\theta = 30^\circ + 180^\circ k\).
• \(\tan \theta = -\frac{1}{\sqrt{3}}\) gives \(\theta = 150^\circ + 180^\circ k\).

For \(0^\circ < \theta < 180^\circ\), the solutions are \(\boxed{\theta = 30^\circ \text{ and } \theta = 150^\circ}\).

Part (i):

Given the equation:

\(\sin(x - 60^\circ) = 3 \cos(x - 45^\circ)\)

Using the angle subtraction identities:

\(\sin(x - 60^\circ) = \sin x \cos 60^\circ - \cos x \sin 60^\circ\)

\(\cos(x - 45^\circ) = \cos x \cos 45^\circ + \sin x \sin 45^\circ\)

Substitute the exact values:

\(\sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2} = 3(\cos x \cdot \frac{\sqrt{2}}{2} + \sin x \cdot \frac{\sqrt{2}}{2})\)

Rearrange and simplify:

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

Combine terms:

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

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

Divide both sides by \(\cos x\):

\(\tan x = \frac{-(6+\sqrt{6})}{(6-\sqrt{2})}\)

Part (ii):

Using the result from part (i), solve for \(x\):

\(x = 118.5^\circ, 298.5^\circ\)