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Math Help - help me verify this identity

  1. #1
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    help me verify this identity

    The original problem

    sin^3x+cos^3x
    ---------------- = 1-sinxcosx
    sinx+cosx

    Now I have this awesome book that tells how to do the problems step by step but it doesn't do a very good job of explaining how it went from step 1 to step 2. For example the book gives me

    sinx^3+cosx^3 _ _ (sinx+cosx)(sinx^2-sinxcosx+cosx^2)
    --------------- = ------------------------------------ = ... = 1-sinxcosx
    sinx+cosx _ _ _ _ _ _ _ _ _ sinx+cosx


    SO my question is how do you get (sinx^2-sinxcosx+cosx^2) from factoring (sinx+cosx) out of (sinx^3+cox^3).


    Oh and the _ _ _ _ _ _ _ _ _ is where spaces would be if the forum didn't cut out multiple spaces
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  2. #2
    Bar0n janvdl's Avatar
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    Quote Originally Posted by omgwtfbbq View Post
    The original problem

    sin^3x+cos^3x
    ---------------- = 1-sinxcosx
    sinx+cosx

    Now I have this awesome book that tells how to do the problems step by step but it doesn't do a very good job of explaining how it went from step 1 to step 2. For example the book gives me

    sinx^3+cosx^3 _ _ (sinx+cosx)(sinx^2-sinxcosx+cosx^2)
    --------------- = ------------------------------------ = ... = 1-sinxcosx
    sinx+cosx _ _ _ _ _ _ _ _ _ sinx+cosx


    SO my question is how do you get (sinx^2-sinxcosx+cosx^2) from factoring (sinx+cosx) out of (sinx^3+cox^3).


    Oh and the _ _ _ _ _ _ _ _ _ is where spaces would be if the forum didn't cut out multiple spaces
    \frac{sin ^{3} x + cos ^{3} x}{sin x + cos x} = 1 - (sinx \ cos x)

    Let sinx = u and cos = v

    LHS:
    \frac{u^3 + v^3}{u + v} = \frac{(u + v)(u^2 - uv + v^2)}{u + v} = u^2 - uv + v^2

    So we have simplified the left side to that.

    u^2 - uv + v^2 = sin ^{2}x + cos ^{2}x - (sinx \ cos x) = 1 - (sinx \ cosx)

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  3. #3
    A riddle wrapped in an enigma
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    Quote Originally Posted by janvdl View Post
    \frac{sin ^{3} x + cos ^{3} x}{sin x + cos x} = 1 - (sinx \ cos x)

    Let sinx = u and cos = v

    LHS:
    \frac{u^3 + v^3}{u + v} = \frac{(u + v)(u^2 - uv + v^2)}{u + v} = u^2 - uv + v^2

    So we have simplified the left side to that.

    u^2 - uv + v^2 = sin ^{2}x + cos ^{2}x - (sinx \ cos x) = 1 - (sinx \ cosx)

    Smoooooothe!!
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