(a) Start with \(\frac{\tan x + \sin x}{\tan x - \sin x} = k\).

Multiply numerator and denominator by \(\cos x\):

\(\frac{\sin x + \sin x \cos x}{\sin x - \sin x \cos x} = k\).

Rearrange to \(\frac{1 + \cos x}{1 - \cos x} = k\).

(b) From \(\frac{1 + \cos x}{1 - \cos x} = k\), cross-multiply to get:

\(k - k \cos x = 1 + \cos x\).

Rearrange to \(k - 1 = (k + 1) \cos x\).

Thus, \(\cos x = \frac{k - 1}{k + 1}\).

(c) Substitute \(k = 4\) into \(\cos x = \frac{k - 1}{k + 1}\):

\(\cos x = \frac{4 - 1}{4 + 1} = \frac{3}{5}\).

Find \(x\) such that \(\cos x = \frac{3}{5}\) within the range \(-\pi < x < \pi\).

The solutions are \(x = \pm 0.927\) (only solutions in the given range).