+284 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!
+118677 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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