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Thread: Help with Complex Number Question (de Moivre's Theorem)

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    Help with Complex Number Question (de Moivre's Theorem)

    The roots of the equation t²-2t+2=0 are a and b. Prove that [(x+a)ⁿ-(x+b)ⁿ]/(a-b) = (sin nθ)/(sinⁿ θ), where cot θ = x +1

    Any help would be appreciated!

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    Re: Help with Complex Number Question (de Moivre's Theorem)



    Note that

    So



    And



    So we have



    So now all you need to prove is



    (Start by finding )

    EDIT: this question was also discussed here: http://community.boredofstudies.org/...arathon-4.html

    EDIT #2: Here's the rest:







    We need to prove



    And since the proof is complete.
    Last edited by fan96; 26 Nov 2018 at 8:13 PM.
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    Re: Help with Complex Number Question (de Moivre's Theorem)

    Assume, without loss of generality (because you can switch a and b and the expression would still be the same)
    a = 1 + i
    b = 1 - i
    (worked out by quadratic formula)
    x + 1 = cot theta
    sub a, b, x+1 basically everything into LHS expression and simplify
    Last edited by Q16slayer; 27 Nov 2018 at 11:32 PM.

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