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Math Help - Prove the following statement(Binomial Theorem)?

  1. #1
    Super Member fardeen_gen's Avatar
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    Prove the following statement(Binomial Theorem)?

    Prove that:
    {(r = 0 to n) nCr cos^(n - r) x sin ^(r) x}{(r = 0 to n) (-1)^(r) nCr cos^(n - r) x sin^(r) x} = cos ^(n) 2x

    NOTE: nCr is n choose r.
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    Prove that:
    {(r = 0 to n) nCr cos^(n - r) x sin ^(r) x}{(r = 0 to n) (-1)^(r) nCr cos^(n - r) x sin^(r) x} = cos ^(n) 2x

    NOTE: nCr is n choose r.
    Remember that (x+y)^n = \sum_{r=0}^n {n\choose r}x^ry^{n-r}.
    Therefore, \sum_{r=0}^n {n\choose r}\cos^{n-r}x\sin^r x = (\cos x + \sin x) \text{ and } \sum_{r=0}^n (-1)^r{n\choose r}\cos^{n-r}x\sin^r x  = (\cos x - \sin x)^n

    If you multiply them you get (\cos^2 x - \sin^2 x)^n = \cos^n (2x).
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