u ; v∈ C such that ∣u∣=∣v∣=1 and uv≠−1. show that ((u+v)/(1+uv)) ∈ R.


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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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