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

Use \(\tan x = \frac{\sin x}{\cos x}\) to rewrite the equation: \(\frac{\frac{\sin x}{\cos x} + \cos x}{\frac{\sin x}{\cos x} - \cos x} = k\).

Clear the fraction: \(\frac{\sin x + \cos^2 x}{\sin x - \cos^2 x} = k\).

Use \(\cos^2 x = 1 - \sin^2 x\) to get \(\frac{\sin x + 1 - \sin^2 x}{\sin x - (1 - \sin^2 x)} = k\).

Simplify: \(\frac{\sin x + 1 - \sin^2 x}{\sin^2 x + \sin x - 1} = k\).

Multiply through by the denominator: \((\sin x + 1 - \sin^2 x) = k(\sin^2 x + \sin x - 1)\).

Expand and simplify: \(\sin x + 1 - \sin^2 x = k\sin^2 x + k\sin x - k\).

Rearrange: \((k + 1)\sin^2 x + (k - 1)\sin x - (k + 1) = 0\).

(b) Solve \(\frac{\tan x + \cos x}{\tan x - \cos x} = 4\).

Using the result from part (a), substitute \(k = 4\): \(5\sin^2 x + 3\sin x - 5 = 0\).

Use the quadratic formula: \(\sin x = \frac{-3 \pm \sqrt{9 + 100}}{10}\).

Calculate: \(\sin x = \frac{-3 \pm 11}{10}\).

Solutions: \(\sin x = 0.8\) or \(\sin x = -1.4\) (discard \(\sin x = -1.4\) as it is not possible).

Find \(x\) for \(\sin x = 0.8\): \(x = 48.1^\circ, 131.9^\circ\).