(c) \( \cos \left(180^{\circ}-4 x\right)+\cos ^{2} 2 x \) \( \begin{array}{ll}\text { (1) } \frac{4 \sin (-4 \theta)}{2 \sin ^{2}\left(360^{\circ}-\theta\right)-1} \\ \text { (3) } \frac{\sin 2 \theta \cos 2 \theta}{} & \text { (4) } \frac{\sin 4 \theta-\sin 2 \theta}{\cos 4 \theta+\cos 2 \theta}\end{array} \)

When examining the expression \( \cos \left(180^{\circ} - 4x\right) + \cos^2 2x \), a deep dive into trigonometric identities can make things clearer. The cosine identity states \( \cos(180^\circ - \theta) = -\cos(\theta) \), which simplifies our expression to \( -\cos(4x) + \cos^2(2x) \). This transformation allows you to see the relationship between these functions more intuitively and is especially useful in solving equations. In practical application, understanding these functions can clarify how waves behave in physics or mechanics. For instance, in oscillating systems like pendulums or springs, both cosine and sine functions model motion and energy effectively. By manipulating these expressions, you can predict the system's behavior over time! So, whether you're solving for angles in a triangle or analyzing sound waves, a good grasp of these identities is your ticket to success.