(a) Start with the given equation:

\(\cos(x - 30^\circ) = 2 \sin(x + 30^\circ)\)

Use the angle subtraction and addition formulas:

\(\cos(x - 30^\circ) = \cos x \cos 30^\circ + \sin x \sin 30^\circ\)

\(\sin(x + 30^\circ) = \sin x \cos 30^\circ + \cos x \sin 30^\circ\)

Substitute the exact values:

\(\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}\)

\(\cos(x - 30^\circ) = \cos x \frac{\sqrt{3}}{2} + \sin x \frac{1}{2}\)

\(\sin(x + 30^\circ) = \sin x \frac{\sqrt{3}}{2} + \cos x \frac{1}{2}\)

Substitute into the original equation:

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

Expand and simplify:

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

Rearrange terms:

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

Factor out terms:

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

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

\(\tan x = \frac{\frac{\sqrt{3}}{2} - 1}{\sqrt{3} - \frac{1}{2}}\)

Simplify to obtain:

\(\tan x = \frac{2 - \sqrt{3}}{1 - 2\sqrt{3}}\)

(b) From part (a), we have:

\(\tan x = \frac{2 - \sqrt{3}}{1 - 2\sqrt{3}}\)

Calculate \(x\) using the inverse tangent function:

\(x = \tan^{-1}\left(\frac{2 - \sqrt{3}}{1 - 2\sqrt{3}}\right)\)

Find solutions within the interval \(0^\circ < x < 360^\circ\):

\(x \approx 173.8^\circ, 353.8^\circ\)