Start with \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}\).

Divide the numerator and the denominator by \(\cos \theta\):

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

Let \(t = \tan \theta = \frac{\sin \theta}{\cos \theta}\), then:

\(\frac{t - 1}{t + 1}\).

This matches the right-hand side of the identity.