Start with the left-hand side (LHS):

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

Expand the expression:

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

Factor out common terms:

\(= \sin \theta(1 - \cos^2 \theta) + \cos \theta(1 - \sin^2 \theta)\)

Use the identity \(1 - \cos^2 \theta = \sin^2 \theta\) and \(1 - \sin^2 \theta = \cos^2 \theta\):

\(= \sin^3 \theta + \cos^3 \theta\)

Thus, the identity is proven: \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta) \equiv \sin^3 \theta + \cos^3 \theta\).