complex numbers involving demoivre

**differentiate** · December 12th 2009, 10:58 PM

show that:

$(1 + cos2\theta + isin\theta )^n + (1+cos\theta - isin\theta )^n = 2^{n+1}cos^n\frac{\theta }{2}cos\frac{n\theta }{2 }$

**pickslides** · December 13th 2009, 12:33 AM

for a complex number $z = x+yi$

then $z^n = r^ncis(n\theta)$ where

$r = \sqrt{x^2+y^2}$ and $\tan(\theta) = \frac{y}{x}$

**Opalg** · December 13th 2009, 12:41 AM

Originally Posted by differentiate

show that:

$(1 + cos{\not2\theta} + isin\theta )^n + (1+cos\theta - isin\theta )^n = 2^{n+1}cos^n\frac{\theta }{2}cos\frac{n\theta }{2 }$ (The 2 on the left should not be there.)

Use the identities $1+\cos\theta=2\cos^2\tfrac\theta2$ and $\sin\theta = 2\sin\tfrac\theta2\cos\tfrac\theta2$ .

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