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Prove that $\left ( 1+ \mathit{i\sqrt{3}} \right )^{\mathit{n}} + \left ( 1- \mathit{i\sqrt{3}} \right )^{\mathit{n}} = 2^{n+1} \cos \; \large\frac{n\pi}{3}$

m, n N.

This is the second part of the multi-part question Q4.

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  • From De moivre's theorem we have
  • (i) $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta,n\in Q$
  • (ii) $(\cos\theta+i\sin\theta)^{-n}=\cos n\theta-i\sin n\theta$
  • (iii) $(\cos\theta-i\sin\theta)^n=\cos n\theta-i\sin n\theta$
  • (iv) $(\sin \theta+i\cos \theta)^n=[\cos(\large\frac{\pi}{2}$$-\theta)+i\sin(\large\frac{\pi}{2}$$-\theta)]^n=\cos n(\large\frac{\pi}{2}$$-\theta)+i\sin n(\large\frac{\pi}{2}$$-\theta)$
  • $e^{i\theta}=\cos\theta+i\sin\theta$
  • $e^{-i\theta}=\cos\theta-i\sin\theta$,also written as $\cos\theta$ and $\cos(-\theta)$
Step 1:
Let $1+\sqrt 3i=r(\cos \theta+i\sin \theta)$
Equating the real and imaginary parts separately
$r\sin\theta=\sqrt 3$
Squaring and adding we get
Step 2:
Also $\alpha=\tan^{-1}\sqrt 3=\large\frac{\pi}{3}$
Since $1+\sqrt 3i$ is represented by a point in quadrant 1.
Therefore $1+\sqrt 3i=2(\cos \large\frac{\pi}{3}$$+i\sin\large\frac{\pi}{3})$
$1-\sqrt 3i=1+\sqrt 3i$
$\qquad\;\;\;\;\;=2(\cos \large\frac{\pi}{3}$$-i\sin\large\frac{\pi}{3})$
Step 3:
Now LHS=$(1+i\sqrt 3)^n+(1-i\sqrt 3)^n$
Hence proved.
answered Jun 11, 2013 by sreemathi.v

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