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    Ammme's Avatar
    Ammme Posts: 1, Reputation: 1
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    #1

    Aug 21, 2010, 07:18 PM
    verify identity (sin^3(x) +cos^3(x)) /sin(x)+cos(x) = 1-sin(x)cos(x)
    I needd helppp. I can't verify this identity! Pleasee help and show full work. Thanks :)
    Unknown008's Avatar
    Unknown008 Posts: 8,076, Reputation: 723
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    #2

    Aug 22, 2010, 02:12 AM

    Ok, you need to find some way to break down the 'cubes' in the cos and sin functions.

    Separate the cos^3 and sin^3 into cos(x)cos^2(x) and sin(x)sin^2(x) respectively:



    Use the identity cos^2(x) + sin^2(x) = 1:



    Expand;



    Rearrange;



    Factor out cos(x)sin(x) from the two last terms in the numerator;



    Factor out sin(x) + cos(x) from the numerator in all the terms together;



    It's done now. :) Only one little step more.
    galactus's Avatar
    galactus Posts: 2,271, Reputation: 282
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    #3

    Aug 22, 2010, 05:11 AM

    If I may show another way.

    Note the sum of two cubes in the numerator. Factoring it, gives:



    The sin(x)+cos(x) cancels and we are left with:



    Note the identity

    This leaves us with


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