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

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

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*}