a) \( 2 \sin \alpha \sin 2 \alpha \sin 3 \alpha \) b) \( 4 \sin 25^{\circ} \cos 15^{\circ} \sin 5^{\circ} \) c) \( 8 \cos 1^{\circ} \cos 2^{\circ} \cos 4^{\circ} \cos 8^{\circ} \) d) \( \cos \alpha \cos (\alpha+2) \) e) \( 8 \cos (\alpha-\beta) \cos (\alpha-\gamma) \cos (\gamma-\beta) \)

The expression \( 2 \sin \alpha \sin 2 \alpha \sin 3 \alpha \) can be simplified using the product-to-sum identities, which can turn products of sine functions into sums or differences, making it easier to integrate, differentiate, or solve equations involving these trigonometric functions. For \( 8 \cos (\alpha-\beta) \cos (\alpha-\gamma) \cos (\gamma-\beta) \), this expression is closely related to the cosine of angles formed by the differences between angles. It's particularly useful in the context of geometry, where these types of cosine products can represent relationships in triangle properties, especially in the formulation of laws such as the Law of Cosines.