Start by finding a common denominator for the left-hand side:

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

Expand the numerator:

\((\cos \theta \sin \theta - \sin^2 \theta) + (\cos \theta + \sin \theta - \cos^2 \theta - \sin \theta \cos \theta)\)

Simplify the numerator:

\(\cos \theta \sin \theta - \sin^2 \theta + \cos \theta + \sin \theta - \cos^2 \theta - \sin \theta \cos \theta\)

Combine like terms:

\(\sin \theta + \cos \theta - \cos^2 \theta - \sin^2 \theta\)

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), the numerator becomes:

\(\sin \theta + \cos \theta - 1\)

The denominator simplifies to:

\(\cos^2 \theta - \sin^2 \theta\)

Using the identity \(\cos^2 \theta - \sin^2 \theta = 1 - 2 \sin^2 \theta\), the expression becomes:

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

This matches the right-hand side, proving the identity.