Show that the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\) can be expressed in the form
$$6 \cos ^ { 2 } x - \cos x - 2 = 0 .$$
Hence solve the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).