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Math Help - trig indentity proof help,

  1. #1
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    trig indentity proof help,

    Deduce that ;

     sin^{4}A + cos^{4}A = \frac{1}{4} (3 + cos4A)

    Can someone start me off please?

    Thanks!

    this is part 'b' of the question, I just realised you might actually need the first part, which I have already done.

    part a is;

    use the identity  sin^{2}A + cos^{2}A = 1 to that show that   sin^{4}A + cos^{4}A = \frac{1}{2}(2-sin^{2}2A)

     (sin^{2}A + cos^{2}A )^{2}= 1

     sin^{4} + cos^{4} + 2sin^{2}A cos^{2}A = 1

     sin^4 + cos^{4} = 1 - 2sin^{2}Acos^{2}A


     = 1-\frac{1}{2} ( 4sin^2Acos^{2}A)  = 1-\frac{1}{2} ( 2sinAcosA)^{2} = 1-\frac{1}{2}sin^{2}2A  =  \frac{1}{2} ( 2-sin^{2}2A)
    Last edited by Tweety; December 15th 2009 at 02:00 PM.
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    I would try

     \sin^2(x) = \frac{1-\cos(2x)}{2}

     \cos^2(x) = \frac{1+\cos(2x)}{2}

    Therefore

     <br />
\sin^4(x)+\cos^4(x) = \left(\frac{1-\cos(2x)}{2}\right)^2+\left(\frac{1+\cos(2x)}{2}\r  ight)^2<br />

    Now expand and group like terms.
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  3. #3
    Lord of certain Rings
    Isomorphism's Avatar
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    Actually you were just one step away from the answer.

    Use this:
    2(2 - \sin^2 2A) = 3 + (1 - 2\sin^2 2A)  = 3 + \cos 4A \implies 2 - \sin^2 2A = \frac12(3 + \cos 4A)
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  4. #4
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    In your part A question you didn't show that :

    sin^{4}A + cos^{4}A =  \frac{1}{2}(2-sin^{2}A)

    You got this solution :

    \frac{1}{2} ( 2-sin^{2}2A)
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  5. #5
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    Quote Originally Posted by ldx2 View Post
    In your part A question you didn't show that :

    sin^{4}A + cos^{4}A =  \frac{1}{2}(2-sin^{2}A)

    You got this solution :

    \frac{1}{2} ( 2-sin^{2}2A)
    oh sorry, its meant to be 2A, I mis-typed
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