Given-that-sinh-1-x-sech-1-x-show-x-5-1-2-

Question Number 178357 by Spillover last updated on 15/Oct/22

Given that \sinh^{- 1} x = sech^{- 1} x

show

x = \sqrt{\frac{\sqrt{5} - 1}{2}}

Answered by Ar Brandon last updated on 16/Oct/22

\sinh^{- 1} x = {sech}^{- 1} x

φ = \sinh^{- 1} x \Rightarrow \sinh φ = x

\cosh^{2} φ - \sinh^{2} φ = 1 \Rightarrow sech φ = \frac{1}{\sqrt{1 + x^{2}}}

\Rightarrow φ = {sech}^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})

\Rightarrow {sech}^{- 1} (\frac{1}{\sqrt{1 + x^{2}}}) = {sech}^{- 1} x

\Rightarrow \frac{1}{\sqrt{1 + x^{2}}} = x \Rightarrow 1 = x^{2} (x^{2} + 1), x > 0

\Rightarrow x^{4} + x^{2} - 1 = 0 \Rightarrow x^{2} = \frac{- 1 + \sqrt{5}}{2}

\Rightarrow x = \sqrt{\frac{\sqrt{5} - 1}{2}}

Commented by Spillover last updated on 16/Oct/22

thank you