Step 1 :Given the trigonometric expression \(\sin (4 x) \cos (2 x)\).
Step 2 :We can simplify it using the product-to-sum identities in trigonometry. The identity we need is \(\sin(a) \cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)]\).
Step 3 :Substitute \(a = 4x\) and \(b = 2x\) into the identity to simplify the expression.
Step 4 :The simplified form of the expression \(\sin (4 x) \cos (2 x)\) is \(\frac{1}{2} \sin (2 x) + \frac{1}{2} \sin (6 x)\).
Step 5 :Final Answer: \(\boxed{\frac{1}{2} \sin (2 x) + \frac{1}{2} \sin (6 x)}\).