I) \( \cos 8 x+\cos 4 x=2 \cos 2 x-4 \operatorname{sen}^{2} 3 x \cdot \cos 2 x \)

A fun fact about trigonometric identities is that they have been used for centuries in various cultures to solve astronomical problems. Ancient Greek astronomers, for instance, relied heavily on these functions to understand planetary movements! They discovered relationships and identities that branched out into modern mathematics, showing us how intertwined math and our understanding of the universe can be. For a more hands-on approach, when tackling equations like this one, it's beneficial to work methodically. Start by rewriting everything in terms of sine and cosine if needed. Common mistakes include overlooking double angles or misapplying identities like \( \sin^2 \theta + \cos^2 \theta = 1 \). Remember, patience and accuracy are key in unwinding these trigonometric puzzles—take your time to check each step!