Answer :
The simplified expression is -(1/2)cos(9x).
None of the provided answer choices match the simplified form.
What is trigonometry?
One of the most significant areas of mathematics, trigonometry has a wide range of applications. The study of how the sides and angles of a right-angle triangle relate to one another is essentially what the field of mathematics known as "trigonometry" is all about.
The expression (cos 3x sin² 3x) can be simplified using trigonometric identities. Let's break it down step by step:
(cos 3x sin² 3x)
Using the identity sin²θ = 1/2 - 1/2cos(2θ), we can rewrite sin² 3x as:
sin² 3x = 1/2 - 1/2cos(2(3x))
= 1/2 - 1/2cos(6x)
Now we can substitute this into the original expression:
(cos 3x sin² 3x) = cos 3x (1/2 - 1/2cos(6x))
Expanding the expression further:
cos 3x (1/2 - 1/2cos(6x)) = (1/2)cos 3x - (1/2)cos 3x cos(6x)
Now, let's simplify each term separately:
(1/2)cos 3x is a standalone term.
Next, we can use the identity cos α cos β = 1/2(cos(α + β) + cos(α - β)) to simplify the second term:
-(1/2)cos 3x cos(6x) = -(1/2)(cos(3x + 6x) + cos(3x - 6x))
= -(1/2)(cos(9x) + cos(-3x))
= -(1/2)(cos(9x) + cos(3x)) (cos(-θ) = cos θ)
Combining both terms:
(1/2)cos 3x - (1/2)cos 3x cos(6x) = (1/2)cos 3x - (1/2)(cos(9x) + cos(3x))
= (1/2)cos 3x - (1/2)cos(9x) - (1/2)cos(3x)
= (1/2)cos 3x - (1/2)cos(3x) - (1/2)cos(9x)
= 0 - (1/2)cos(9x)
= -(1/2)cos(9x)
Therefore, the simplified expression is -(1/2)cos(9x).
None of the provided answer choices match the simplified form.
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