Problem

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

Answer

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Answer

Final Answer: \(\boxed{\frac{1}{2} \sin (2 x) + \frac{1}{2} \sin (6 x)}\).

Steps

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: \(\boxed{\frac{1}{2} \sin (2 x) + \frac{1}{2} \sin (6 x)}\).

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