Start with the left-hand side (LHS):

\(\frac{\sin \theta}{1 - \cos \theta} - \frac{1}{\sin \theta}\)

Put over a common denominator:

\(\frac{\sin^2 \theta - (1 - \cos \theta)}{(1 - \cos \theta) \sin \theta}\)

Simplify the numerator using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\):

\(\frac{1 - \cos^2 \theta - 1 + \cos \theta}{(1 - \cos \theta) \sin \theta}\)

\(\frac{\cos \theta (1 - \cos \theta)}{(1 - \cos \theta) \sin \theta}\)

Cancel \(1 - \cos \theta\) from the numerator and denominator:

\(\frac{\cos \theta}{\sin \theta}\)

Recognize this as \(\frac{1}{\tan \theta}\).