hi i just started class and my sports competition was this weekend so I would really appreciate it if someone could guide me through this

The complex numbers $z$ and $w$ satisfy $|z| = |w| = 1$ and $zw \ne -1.$

(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.

even my tutor couldn't figure out how to get around this....

thank you in ad vance :D

Guest Feb 26, 2022

#2**0 **

The complex numbers \(z\) and \(w\) satisfy \(|z| = |w| = 1\) and \(zw \ne -1.\)

(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.

sorry about that :(

Guest Feb 26, 2022

#3**+1 **

The complex numbers $z$ and $w$ satisfy $|z| = |w| = 1$ and $zw \ne -1.$

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

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

\(Let \quad z=e^{i\theta}=cos\theta+ isin \theta\qquad\\ and \quad w=e^{i\alpha}=cos\alpha +isin \alpha\\ \theta+\alpha \ne (2n+1)\pi\\~\\ Prove \;\; \bar z=\frac{1}{z} \\\bar z=cos\theta-isin\theta\\ \frac{1}{z}=(e^{i\theta})^{-1}\\ \frac{1}{z}=e^{-\theta i}\\ \frac{1}{z}=cos(-\theta)+isin(-\theta)\\ \frac{1}{z}=cos(\theta)-isin(\theta)\\ \frac{1}{z}=\bar z \qquad QED \)

Melody Feb 26, 2022