solve-tanh-x-1-cosh-x-

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

solve \tanh (x) = \frac{1}{\cosh (x)}

Answered by MJS last updated on 25/Mar/20

\frac{\sinh x}{\cosh x} = \frac{1}{\cosh x}

\cosh x \neq 0 \forall x \in R

\cosh x \neq 0 \Rightarrow x \neq (\frac{2 n + 1}{2}) π i

\sinh x = 1

\Rightarrow x = \ln (1 + \sqrt{2}) \in R

x = \ln (1 + \sqrt{2}) + 2 n π i \lor x = \ln (- 1 + \sqrt{2}) + (2 n + 1) π i

Answered by MJS last updated on 25/Mar/20

let x = 2 arctanh t \Leftrightarrow t = \tanh \frac{x}{2}

\frac{2 t}{t^{2} + 1} = - \frac{t^{2} - 1}{t^{2} + 1}

\Rightarrow t = - 1 \pm \sqrt{2}

\Rightarrow x = \ln (1 + \sqrt{2}) + 2 n π i \lor x = \ln (- 1 + \sqrt{2}) + (2 n + 1) π i

Commented by jagoll last updated on 25/Mar/20

thank you mr

Facebook Tweet Pin