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Question Number 206160 by MATHEMATICSAM last updated on 08/Apr/24

$$\mathrm{If}x\mathrm{and}y\mathrm{are}\mathrm{real}\mathrm{numbers}\mathrm{then}\mathrm{is}\mathrm{it}$$$$\mathrm{possible}\mathrm{that}\sec \theta =\frac{2xy}{x^2+y^2}?$$

Commented by mr W last updated on 08/Apr/24

$$onlyifx=\pm y.$$

Answered by mahdipoor last updated on 08/Apr/24

$$seca=\frac{2xy}{x^2+y^2}\in ]-1,1[\Rightarrow$$$$\frac{2xy}{x^2+y^2}⩽-1\Rightarrow 2xy⩽-x^2-y^2\Rightarrow {(x+y)}^2⩽0$$$$\Rightarrow x=-y$$$$\frac{2xy}{x^2+y^2}⩾1\Rightarrow 2xy⩾x^2+y^2\Rightarrow 0⩾{(x-y)}^2$$$$\Rightarrow x=y$$$$\Rightarrow \Rightarrow x=\pm y$$

Commented by Frix last updated on 08/Apr/24

$$\sec \alpha =\frac{1}{\cos \alpha }\Rightarrow \sec \alpha \in (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$$

Answered by Frix last updated on 08/Apr/24

$$\sec \theta =\frac{1}{\cos \theta }$$$$\cos \theta =\frac{x^2+y^2}{2xy}=c$$$$x^2+y^2=2cxy$$$$y^2-2cxy+x^2=0$$$$y=(c\pm \sqrt{c^2-1})x$$$$y=(\cos \theta \pm \mathrm{i}\sin \theta )x$$$$y\in \mathbb{R}\Rightarrow \theta =n\pi \Rightarrow \cos \theta =\pm 1\Rightarrow y=\pm x$$

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