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Hi, could someone help me with this question?

 

The complex numbers z and w satisfy |z| = |w| = 1 and zw =/= -1.

(a) Prove that \(\overline{z} = \frac{1}{z}\) and \(\overline{w} = \frac{1}{w}\).

(b) Prove that \(\frac{z + w}{zw + 1}\) is a real number. 

 

Thank you so much!

 Jan 14, 2021
 #1
avatar+112032 
+2

Part a is easy.

 

Let

  \(z=e^{i\theta}=cos\theta + isin \theta\\ \bar{z}=cos\theta - isin \theta\\ \\~\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\times \frac{cos\theta - isin \theta}{cos\theta - isin \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{cos^2\theta + sin^2 \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{1}\\ \frac{1}{z}=cos\theta - isin \theta\\ \frac{1}{z}=\bar z\)

 

Same logic for w 

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 Jan 14, 2021
edited by Melody  Jan 17, 2021
 #2
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Thank you, Melody!

Caffeine  Jan 15, 2021
 #3
avatar+112032 
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I would like to see someone answer this too.   indecision

 Jan 15, 2021
 #4
avatar+31514 
+3

Part b can be done as follows:

 

Alan  Jan 15, 2021
 #5
avatar+112032 
0

Thanks Alan,

That makes sense.  laugh

Melody  Jan 15, 2021

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