$\cos (2 x) \cos (4 x)$

Step 1 ：The problem is asking for the product of \(\cos (2 x)\) and \(\cos (4 x)\). This is a straightforward multiplication of two trigonometric functions.

Step 2 ：We can simplify this expression using the product-to-sum identities in trigonometry. The product-to-sum identities are given by: \(\cos a \cos b = \frac{1}{2} [\cos(a - b) + \cos(a + b)]\).

Step 3 ：We can use this identity to simplify the expression \(\cos (2 x) \cos (4 x)\).

Step 4 ：Applying the product-to-sum identity, the expression \(\cos (2 x) \cos (4 x)\) simplifies to \(\frac{1}{2} \cos (2 x) + \frac{1}{2} \cos (6 x)\).

Step 5 ：Final Answer: The simplified form of \(\cos (2 x) \cos (4 x)\) is \(\frac{1}{2} \cos (2 x) + \frac{1}{2} \cos (6 x)\). Therefore, the final answer is \(\boxed{\frac{1}{2} \cos (2 x) + \frac{1}{2} \cos (6 x)}\).