SOLUTION: Use a sum or difference formula to write a formula for cos⁡2x. Express the result in terms of sin⁡x and cos⁡x.

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Question 1037392: Use a sum or difference formula to write a formula for cos⁡2x. Express the result in terms of sin⁡x and cos⁡x.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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1. Use a sum or difference formula to write a formula for cos(2x).
2. Express the result in terms of sin(x) and cos(x).
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1.  The "addition formula" for cosine is 

    cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)

    (see the lesson Addition and subtraction formulas in this site.)


    Take x = y in this formula, and you will get 

    cos(2x) = cos%5E2%28x%29+-+sin%5E2%28x%29.   (1)

    It is the answer for #1.


2.  You can express (1) in terms of sin(x) only.
    For it, substitute  cos^2(x) = 1 - sin^2(x)  into (1).  You will get

    cos(2x) = 1+-+2%2Asin%5E2%28x%29.   (2)


    Or,  you can express (1) in terms of cos(x) only.
    For it, substitute  sin^2(x) = 1 - cos^2(x)  into (1).  You will get

    cos(2x) = 2%2Acos%5E2%28x%29+-+1.   (3)