Show that the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = k\), where \(k\) is a constant, can be expressed as
$$( k + 1 ) \sin ^ { 2 } x + ( k - 1 ) \sin x - ( k + 1 ) = 0$$
Hence solve the equation \(\frac { \tan x + \cos x } { \tan x - \cos x } = 4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).