Show that the equation
$$\frac { \tan x + \sin x } { \tan x - \sin x } = k$$
where \(k\) is a constant, may be expressed as
$$\frac { 1 + \cos x } { 1 - \cos x } = k$$
Hence express \(\cos x\) in terms of \(k\).
Hence solve the equation \(\frac { \tan x + \sin x } { \tan x - \sin x } = 4\) for \(- \pi < x < \pi\).