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Math Help - Teach me the best way of simplyfying this Long Demoivre's Trig Prooving question.

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    Teach me the best way of simplyfying this Long Demoivre's Trig Prooving question.

    Hi there,i've attached the question in the image...It's basically some test question that takes my time, just being curious can one simplify my technique? thank you very much.
    Attached Thumbnails Attached Thumbnails Teach me the best way of simplyfying this Long Demoivre's Trig Prooving question.-dsc00194.jpg  
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    For a start, the identities should be

    z-\frac1z = 2i\sin\theta,

    z^n - z^{-n} = 2i\sin(n\theta).

    You will also need the identity z^n + z^{-n} = 2\cos(n\theta).

    Then you can form the binomial expansion

    \begin{aligned}(2i\sin\theta)^6 = (z-z^{-1})^6 &= z^6 -6z^4 + 15z^2 - 20 + 15z^{-2} - 6z^{-4} + z^{-6}\\ &= (z^6+ z^{-6}) - 6(z^4 + z^{-4}) + 15(z^2 + z^{-2}) - 20\\ &= 2\cos(6\theta) - 12\cos(4\theta) + 30\cos(2\theta) -20,\end{aligned}

    and the result follows easily.
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    Thank you so much
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    Quote Originally Posted by afterahardtry View Post
    Hi there,i've attached the question in the image...It's basically some test question that takes my time, just being curious can one simplify my technique? thank you very much.
    Let z=r\mathrm{e}^{i\theta} for some r\in[0,\infty) and \theta\in[0,2\pi).
    Then, we have z=r(\cos(\theta)+i\sin(\theta)).
    For n\in\mathbb{N}, we clearly have
    [r(\cos(\theta)+i\sin(\theta))]^{n}=z^{n}
    ............................._ =[r\mathrm{e}^{i\theta}]^{n}
    ............................._ =r^{n}\mathrm{e}^{in\theta}
    ............................._ =r^{n}(\cos(n\theta)+i\sin(n\theta)).
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    Quote Originally Posted by Opalg View Post

    Then you can form the binomial expansion

    \begin{aligned}(2i\sin\theta)^6 = (z-z^{-1})^6 &= z^6 -6z^4 + 15z^2 - 20 + 15z^{-2} - 6z^{-4} + z^{-6}\\ &= (z^6+ z^{-6}) - 6(z^4 + z^{-4}) + 15(z^2 + z^{-2}) - 20\\ &= 2\cos(6\theta) - 12\cos(4\theta) + 30\cos(2\theta) -20,\end{aligned}

    and the result follows easily.

    Dear sir, for some reasons i would have a negative integer ,example
    1/32[ xxxx - 10] ,
    in oppose of the proove where it is seeking for
    1/32[ xxxx + 10] ,
    from what i have learn -sin (theta) = - sin (theta) , and -cos (theta) = cos ( theta) which i could not apply in this theorem, is there a way for changing the the proove without jeopardizing sin^6 into negative? thank you very much for the help .
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    Quote Originally Posted by afterahardtry View Post
    Dear sir, for some reasons i would have a negative integer ,example
    1/32[ xxxx - 10] ,
    in oppose of the proove where it is seeking for
    1/32[ xxxx + 10] ,
    from what i have learn -sin (theta) = - sin (theta) , and -cos (theta) = cos ( theta) which i could not apply in this theorem, is there a way for changing the the proove without jeopardizing sin^6 into negative? thank you very much for the help .
    I don't know whether this is the cause of the dificulty, but I wonder if you are overlooking the "i" in (2i\sin\theta)^6. Since i^6=-1, this expression becomes -64\sin^6\theta.
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