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Math Help - de moivre theorem and summation

  1. #1
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    de moivre theorem and summation

    By considering \sum(1+\iota \tan \theta)^k from k=0 to k=n-1, show that
    \sum \cos k\theta \sec^k \theta = \cot \theta\sin n\theta\sec^n\theta

    stuck here
    please help

    and i also dont know how to put limits in summation :P please tell that too
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  2. #2
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    Re: de moivre theorem and summation

    Start by writing (1+i\tan \theta)^{k} as

    \left(1+i \frac{\sin \theta}{\cos \theta}\right)^{k}=\frac{(\cos \theta + i \sin \theta)^{k}}{\cos^{k}\theta},

    and then make use of De Moivre's theorem.
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  3. #3
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    Re: de moivre theorem and summation

    i already did what you told but cant go any further
    can you please continue after that?
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  4. #4
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    Re: de moivre theorem and summation

    If you do as suggested the summation can be split into two parts, one real the other imaginary.
    The summation itself can be summed easily since it is a GP with common ratio (1+i\tan \theta).
    Do that and then equate reals and imaginaries across the two results.
    (Actually, you only need to equate reals.)
    Last edited by BobP; April 24th 2013 at 08:23 AM.
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