u-v-C-such-that-u-v-1-and-uv-1-show-that-u-v-1-uv-R-

Question Number 126552 by mathocean1 last updated on 21/Dec/20

u; v \in C s u c h t h a t ∣ u ∣=∣ v ∣= 1

a n d u v \neq - 1 .

s h o w t h a t \frac{u + v}{1 + u v} \in R .

Answered by Olaf last updated on 21/Dec/20

u = e^{i α}, v = e^{i β}

\frac{u + v}{1 + u v} = \frac{e^{i α} + e^{i β}}{1 + e^{i α} e^{i β}} = \frac{e^{i \frac{α + β}{2}} (e^{i \frac{α - β}{2}} + e^{- i \frac{α - β}{2}})}{e^{i \frac{α + β}{2}} (e^{i \frac{α + β}{2}} + e^{- i \frac{α + β}{2}})}

= \frac{\cos (\frac{α - β}{2})}{\cos (\frac{α + β}{2})} \in R

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