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Question Number 113451 by pete last updated on 13/Sep/20

$$\mathrm{If}2\mathrm{x}={\mathrm{a}}^{\mathrm{n}}+{\mathrm{a}}^{-\mathrm{n}}\mathrm{and}2\mathrm{y}={\mathrm{a}}^{\mathrm{n}}-{\mathrm{a}}^{-\mathrm{n}}\mathrm{calculate}$$$$\mathrm{the}\mathrm{value}\mathrm{of}{\mathrm{x}}^2-{\mathrm{y}}^2\mathrm{in}\mathrm{its}\mathrm{simplest}\mathrm{form}$$

Answered by bemath last updated on 13/Sep/20

$$4x^2=a^{2n}+a^{-2n}+2$$$$4y^2=a^{2n}+a^{-2n}-2$$$$\Rightarrow 4x^2-4y^2=4$$$$\Rightarrow x^2-y^2=1$$

Answered by Her_Majesty last updated on 13/Sep/20

$$2x=2cosh\left(nlna\right)$$$$2y=2sinh\left(nlna\right)$$$$x^2-y^2=1$$

Commented by Her_Majesty last updated on 13/Sep/20

$${you}^{\prime }rewrong.$$$$2x=a^n+a^{-n}=e^{nlna}+e^{-nlna}=2\times (e^{nlna}+e^{-nlna})/2=$$$$=2cosh\left(nlna\right)$$$$similar2y=2sinh\left(nlna\right)$$$${cosh}^2\theta -{sinh}^2\theta =1$$

Answered by MJS_new last updated on 13/Sep/20

$$\frac{{(u+v)}^2}{4}-\frac{{(u-v)}^2}{4}=uv$$$$herev=\frac{1}{u}\Rightarrow uv=1$$

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