Start with the given equation:

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

Express \(\tan x\) as \(\frac{\sin x}{\cos x}\):

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

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

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

Factor out \(\sin x\) from both numerator and denominator:

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

Cancel \(\sin x\) from numerator and denominator:

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