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Question Number 85845 by jagoll last updated on 25/Mar/20

$$\mathrm{solve}\tanh \left(\mathrm{x}\right)=\frac{1}{\cosh \left(\mathrm{x}\right)}$$

Answered by MJS last updated on 25/Mar/20

$$\frac{\sinh x}{\cosh x}=\frac{1}{\cosh x}$$$$\cosh x\ne 0\mathrm{\forall }x\in \mathbb{R}$$$$\cosh x\ne 0\Rightarrow x\ne \left(\frac{2n+1}{2}\right)\pi \mathrm{i}$$$$\sinh x=1$$$$\Rightarrow x=\ln (1+\sqrt{2})\in \mathbb{R}$$$$x=\ln (1+\sqrt{2})+2n\pi \mathrm{i}\vee x=\ln (-1+\sqrt{2})+(2\mathrm{n}+1)\pi \mathrm{i}$$

Answered by MJS last updated on 25/Mar/20

$$\mathrm{let}x=2\mathrm{arctanh}t\iff t=\tanh \frac{x}{2}$$$$\frac{2t}{t^2+1}=-\frac{t^2-1}{t^2+1}$$$$\Rightarrow t=-1\pm \sqrt{2}$$$$\Rightarrow x=\ln (1+\sqrt{2})+2n\pi \mathrm{i}\vee x=\ln (-1+\sqrt{2})+(2n+1)\pi \mathrm{i}$$

Commented by jagoll last updated on 25/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{mr}$$

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