Start with the left-hand side: \((\sin x + \cos x)(1 - \sin x \cos x)\).

Expand the expression: \(\sin x + \cos x - \sin^2 x \cos x - \cos^2 x \sin x\).

Rearrange terms: \(\sin x + \cos x - \sin^2 x \cos x - \cos^2 x \sin x\).

Use the identities \(\sin^2 x = 1 - \cos^2 x\) and \(\cos^2 x = 1 - \sin^2 x\).

Substitute these into the expression: \(\sin x + \cos x - (1 - \cos^2 x) \cos x - (1 - \sin^2 x) \sin x\).

Simplify: \(\sin x + \cos x - \cos x + \cos^3 x - \sin x + \sin^3 x\).

Combine like terms: \(\sin^3 x + \cos^3 x\).

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