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Question Number 178575 by Spillover last updated on 18/Oct/22

$$\mathrm{Solve}\mathrm{for}\mathrm{x}$$$${\mathrm{e}}^{\sinh {}^{-1}\mathrm{x}}=1+{\mathrm{e}}^{\cosh {}^{-1}\mathrm{x}}$$

Answered by mr W last updated on 19/Oct/22

$$x+\sqrt{x^2+1}=1+x+\sqrt{x^2-1}$$$$\sqrt{x^2+1}=1+\sqrt{x^2-1}$$$$x^2+1=x^2+2\sqrt{x^2-1}$$$$1=2\sqrt{x^2-1}$$$$x^2-1=\frac{1}{4}$$$$\Rightarrow x=\pm \frac{\sqrt{5}}{2}$$

Commented by Spillover last updated on 19/Oct/22

$$\mathrm{thanks}$$

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