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    I had a problem in an Fp2 paper I did last night

    Show that \cos 6 \theta \equiv 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta -1

    So I noted that \cos 6 \theta + i\sin 6 \theta = (cos \theta + isin \theta)^6 and used binomial expansion twice to do the thing. It was a trawl of algebra but straight-forward. But my writing for one line actually ended up spanning two (from the MS, I see I could have used c for \cos \theta and s for \sin \theta).

    My working

    Using x instead of \theta for the sake of simplicity.
    \begin{aligned} \cos6x & =\cos^6x-15\cos^4x\sin^2x+15\cos^2x\sin^4  x-\sin^6x  \\ & =\cos^5x-15\cos^4x+15\cos^5x+15\cos^2x(1-2\cos^2x+\cos^4x)-\sin^6x \\ &=\cos^x-15\cos^4x+15\cos^6x+15\cos^2x-30\cos^4x+15\cos^4x-\sin^x \\ &=31\cos^6x-45\cos^4x+15\cos^2x-(1-\cos^2x)^3 \\ &=31\cos^6x-45\cos^4x+15\cos^2x+\cos^6x-3\cos^4x+3\cos^2x-1 \\ &=32\cos^6x-48\cos^4x+18\cos^2x-1  \\ & \text{As required}\end{aligned}


    In previous years on here, I've seen a method which avoided writing so many terms by considering only the real or imaginary coefficients from the getgo.

    It used \displaystyle \sum and \Re or \Im as appropriate but I don't recall how it was notated. Can someone show me below?
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    (Original post by Kvothe the arcane)
    It used \displaystyle \sum and \Re or \Im as appropriate but I don't recall how it was notated. Can someone show me below?
    Something like:

    Writing \cos n\theta +i\sin n\theta =  (c+is)^n by De Moivre.

    Then,

    \displaystyle\cos 6\theta = \Re (c+is)^6\\=\Re\sum_{l=0}^6\binom  {n}{l}c^l(is)^{6-l}\\=\sum^6_{\substack{l=0\\ \text{l even}}}\binom{n}{l}c^l(is)^{6-l}

    \displaystyle=\binom{6}{0}c^0(is  )^6+\binom{6}{2}c^2(is)^4+\binom  {6}{4}c^4(is)^2+\binom{6}{6}c^6(  is)^0

    then as you had.
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    ghostwalker, thanks .

    Nice latex and use of \substack{}
    Similar but odd for powers of sin I take it?
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    (Original post by Kvothe the arcane)

    Similar but odd for powers of sin I take it?
    And use \Im and not \Re.
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    (Original post by Zacken)
    And use \Im and not \Re.
    Naturally :awesome:.
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    (Original post by Kvothe the arcane)
    ghostwalker, thanks .
    np

    Nice latex and use of \substack{}
    Had to look that one up

    Similar but odd for powers of sin I take it?
    Aye.
 
 
 
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