verify-that-cosh-cosh-1-y-y-if-y-1-

Question Number 82953 by TawaTawa1 last updated on 26/Feb/20

verify that : \cosh . \cosh^{- 1} (y) = y, if y \in (1, + \infty)

Commented by mathmax by abdo last updated on 26/Feb/20

w e h a v e {c h}^{- 1} (y) = l n (y + \sqrt{y^{2} - 1}) \Rightarrow

c h ({c h}^{- 1} (y) = c h (l n (y + \sqrt{y^{2} - 1})

= \frac{e^{l n (y + \sqrt{y^{2} - 1})} + e^{- l n (y + \sqrt{y^{2} - 1})}}{2} = \frac{y + \sqrt{y^{2} - 1 +} \frac{1}{y + \sqrt{y^{2} - 1}}}{2}

= \frac{{(y + \sqrt{y^{2} - 1})}^{2} + 1}{2 (y + \sqrt{y^{2} - 1})} = \frac{y^{2} + 2 y \sqrt{y^{2} - 1} + y^{2} - 1 + 1}{2 (y + \sqrt{y^{2} - 1})}

= \frac{2 y^{2} + 2 y \sqrt{y^{2} - 1}}{2 (y + \sqrt{y^{2} - 1})} = \frac{y^{2} + y \sqrt{y^{2} - 1}}{y + \sqrt{y^{2} - 1}} = \frac{y (y + \sqrt{y^{2} - 1})}{y + \sqrt{y^{2} - 1}} = y

Commented by mathmax by abdo last updated on 26/Feb/20

y o u a r e w e l c o m e

Commented by TawaTawa1 last updated on 26/Feb/20

God bless you sir