Can we find the value of $\sin 18^\circ$ as a surd?

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Show that \(\theta=18^\circ\) satisfies the equation \(\cos 3\theta=\sin 2\theta\). By expressing \(\cos 3\theta\) and \(\sin 2\theta\) in terms of trigonometrical functions of \(\theta\), show that \(\sin 18^\circ\) is a root of the equation \[4x^2+2x-1=0,\] and hence find its value as a surd.

Find all solutions between \(0^\circ\) and \(360^\circ\) of the equation \[4\tan 2x=\cot x.\]