1. **State the problem:** Prove the trigonometric identity $$1 - \cos 5\theta \cos 3\theta - \sin 5\theta \sin 3\theta = 2 \sin^2 \theta$$. 2. **Recall the cosine addition formula:** $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$. 3. **Apply the formula to the left side:** $$\cos 5\theta \cos 3\theta + \sin 5\theta \sin 3\theta = \cos(5\theta - 3\theta) = \cos 2\theta$$. 4. **Rewrite the left side of the identity:** $$1 - \cos 5\theta \cos 3\theta - \sin 5\theta \sin 3\theta = 1 - \cos 2\theta$$. 5. **Use the double-angle identity for cosine:** $$\cos 2\theta = 1 - 2 \sin^2 \theta$$. 6. **Substitute into the expression:** $$1 - \cos 2\theta = 1 - (1 - 2 \sin^2 \theta) = 2 \sin^2 \theta$$. 7. **Conclusion:** The left side simplifies exactly to the right side, so the identity is proven. **Final answer:** $$1 - \cos 5\theta \cos 3\theta - \sin 5\theta \sin 3\theta = 2 \sin^2 \theta$$.