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

Question Number 190935 by Spillover last updated on 14/Apr/23

G i v e n t h a t \sinh^{- 1} y = sech^{- 1} y

s h o w t h a t y^{2} = \frac{\sqrt{5} - 1}{2}

Answered by mehdee42 last updated on 14/Apr/23

{s i n h}^{- 1} y = a \Rightarrow y = s i n h a = \frac{e^{a} - e^{- a}}{2}

{s e c h}^{- 1} y = a \Rightarrow y = s e c h a = \frac{2}{e^{a} + e^{- a}}

\frac{e^{a} - e^{- a}}{2} = \frac{2}{e^{a} + e^{- a}} \Rightarrow e^{2 a} - e^{- 2 a} = 4

i f e^{2 a} = u \Rightarrow u^{2} - u - 1 = 0 \Rightarrow u = \frac{1 + \sqrt{5}}{2} = e^{2 a}

y^{2} = \frac{e^{2 a} + e^{- 2 a} - 2}{4} = \frac{\frac{1 + \sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2} - 2}{4} = \frac{\sqrt{5} - 1}{2}

Commented by Spillover last updated on 15/Apr/23

thank you