De Moivre's Theorem
As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem:
![$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta)
$](http://www.dsprelated.com/josimages_new/mdft/img227.png)
![$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\left[e^{j\theta}\right] ^n = e^{j\theta n} =
\cos(n\theta) + j \sin(n\theta)
$](http://www.dsprelated.com/josimages_new/mdft/img228.png)

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Conclusion
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Euler's Identity
As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem: