SOLUTION: I need help with these trigonometry problems... the question says "Write the following as a single trig expression." 1. 2 sin4x cos 4x 2. 2 cos(squared)3x -1 3. 1-2 sin(squared)

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Question 614139: I need help with these trigonometry problems... the question says "Write the following as a single trig expression."
1. 2 sin4x cos 4x
2. 2 cos(squared)3x -1
3. 1-2 sin(squared)4x

Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
These problems are meant to help you learn that the Trig properties/formulas/identities you've learned are patterns. The x's, A's and B's in the various formulas should be looked upon as placeholders. They can be replaced by any number, variable or expression and the equation will still be true!

So when you look at
sin(2x) = 2sin(x)cos(x)
think "the sin of 2 times anything is equal to 2 times the sin of the anything times the cos of the anything (and vice versa)" Once you get comfortable with this you will start seeing that your problem #1:
2sin(4x)cos(4x)
exactly matches the pattern of the right side of sin(2x) with the "anything" being 4x. So according to the pattern this must be equal to the sin of 2 times the anything:
sin(2*(4x)) or sin(8x)

#2.
This fits the pattern of the right side of:

with the "anything" being 3x. So according to the pattern the right side must be equal to the left side: the cos of 2 times the "anything":
cos(2(3x)) = cos(6x)

#3
This fits the pattern of the right side of:

In hopes that you've started understanding this, I'll leave it up to you to finish.