Step-by-step Solution:

\begin{aligned} &\text{We use the cosine difference identity:} \\ &\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right). \\ \\ &\text{Step 1: Identify } A \text{ and } B \text{ in our case:} \\ &A = 7x, \quad B = 5x. \\ \\ &\text{Applying the identity:} \\ &\cos 5x - \cos 7x = -2 \sin\left(\frac{7x + 5x}{2}\right) \sin\left(\frac{7x - 5x}{2}\right). \\ \\ &\text{Step 2: Simplify the expression:} \\ &\cos 5x - \cos 7x = -2 \sin\left(\frac{12x}{2}\right) \sin\left(\frac{2x}{2}\right). \\ \\ &= -2 \sin(6x) \sin(x). \\ \\ &\text{Final Answer:} \quad \cos 5x - \cos 7x = -2 \sin(6x) \sin(x). \end{aligned}