1. The problem is to simplify the expression $\sin 2\theta - \sin^2 \frac{\theta}{2}$.\n\n2. Recall the double-angle identity for sine: $\sin 2\theta = 2 \sin \theta \cos \theta$. Also, remember the half-angle identity: $\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}$.\n\n3. Substitute these identities into the expression:\n$$\sin 2\theta - \sin^2 \frac{\theta}{2} = 2 \sin \theta \cos \theta - \frac{1 - \cos \theta}{2}$$\n\n4. To combine terms, write the expression with a common denominator 2:\n$$= \frac{4 \sin \theta \cos \theta}{2} - \frac{1 - \cos \theta}{2} = \frac{4 \sin \theta \cos \theta - (1 - \cos \theta)}{2}$$\n\n5. Simplify the numerator:\n$$4 \sin \theta \cos \theta - 1 + \cos \theta = 4 \sin \theta \cos \theta + \cos \theta - 1$$\n\n6. Factor $\cos \theta$ from the first two terms:\n$$= \cos \theta (4 \sin \theta + 1) - 1$$\n\n7. So the simplified expression is:\n$$\frac{\cos \theta (4 \sin \theta + 1) - 1}{2}$$\n\nThis is the simplified form of the original expression.\n\nFinal answer: $$\boxed{\frac{\cos \theta (4 \sin \theta + 1) - 1}{2}}$$