+0

# hi help pls :)) ty

0
83
3

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

Feb 26, 2022

#1
+117224
0

Without unrendered LaTex

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.

Feb 26, 2022
#3
+117224
+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.

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

Feb 26, 2022