Answer
$$\sinh \left(\cosh ^{-1} t\right)=\sqrt{t^{2}-1} \quad \text { for } t \geq 1$$
Work Step by Step
Let $ x=\cosh^{-1}t\ \to \ t=\cosh x$, then
\begin{align*}
\sinh^2 \left(\cosh ^{-1} t\right)&= \sinh^2x\\
&= \cosh^2 x-1\\
\sinh \left(\cosh ^{-1} t\right)&= \sqrt{\cosh^2 x-1}\\
&=\sqrt{t^2-1},\ \ \ \ \ t\geq 1
\end{align*}
Hence, verified.
