if-1-x-2-x-1-y-2-y-1-with-x-y-R-find-x-y-2-

Question Number 189022 by mr W last updated on 10/Mar/23

$${if}\:\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }+{x}\right)\left(\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }+{y}\right)=\mathrm{1}\: \\ $$$${with}\:{x},{y}\:\in{R},\:{find}\:\left({x}+{y}\right)^{\mathrm{2}} =? \\ $$

Answered by Frix last updated on 11/Mar/23

$$\mathrm{Obviously}\:{y}=−{x}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{0} \\ $$$$\left[\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+{x}=\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}−{x}}\right] \\ $$

Commented by mr W last updated on 11/Mar/23

$${can}\:{we}\:{be}\:{sure}\:{that}\:{no}\:{other}\:{solutions} \\ $$$${exist}? \\ $$

Answered by Frix last updated on 11/Mar/23

$$\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+{x}=\frac{\mathrm{1}}{\:\sqrt{{y}^{\mathrm{2}} +\mathrm{1}}+{y}} \\ $$$$\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+{x}=\sqrt{{y}^{\mathrm{2}} +\mathrm{1}}−{y} \\ $$$${x}+{y}=\sqrt{{y}^{\mathrm{2}} +\mathrm{1}}−\sqrt{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$${x}^{\mathrm{2}} +\mathrm{2}{xy}+{y}^{\mathrm{2}} ={y}^{\mathrm{2}} +\mathrm{1}+{x}^{\mathrm{2}} +\mathrm{1}−\mathrm{2}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\sqrt{{y}^{\mathrm{2}} +\mathrm{1}} \\ $$$${xy}=\mathrm{1}−\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\sqrt{{y}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\mathrm{1}−{xy}=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\sqrt{{y}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\mathrm{1}−\mathrm{2}{xy}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} ={x}^{\mathrm{2}} {y}^{\mathrm{2}} +{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1} \\ $$$$−\mathrm{2}{xy}={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} =\mathrm{0} \\ $$

Commented by mr W last updated on 11/Mar/23

$${nice}\:{solution},\:{thanks}! \\ $$

Answered by MJS_new last updated on 11/Mar/23

$${z}=\mathrm{sinh}\:{u}\:\wedge{y}=\mathrm{sinh}\:{v} \\ $$$$\mathrm{e}^{{u}} \mathrm{e}^{{v}} =\mathrm{1} \\ $$$$\mathrm{e}^{{u}+{v}} =\mathrm{1} \\ $$$${u}+{v}=\mathrm{0} \\ $$$${v}=−{u} \\ $$$$\mathrm{sinh}\:\left(−{u}\right)\:=−\mathrm{sinh}\:{u} \\ $$$${y}=−{x} \\ $$

Commented by mr W last updated on 11/Mar/23

$${nice}\:{solution},\:{thanks}! \\ $$

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