Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.9 Hyperbolic Functions - Exercises - Page 384: 49


$$\cosh \left(\sinh ^{-1} t\right)=\sqrt{t^{2}+1}$$

Work Step by Step

Let $x= \sinh^{-1}t\ \ \to \ \ \sinh x=t $ Then we have: \begin{align*} \cosh ^2\left(\sinh ^{-1} t\right)&=\cosh^2 x\\ &= 1+\sinh^2x\\ \cosh x&=\sqrt{1+\sinh^2 x}\\ \cosh (\sinh^{-1}t)&= \sqrt{1+t^2 } \end{align*}
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