Answer
$\begin{align*}
(a)& \quad 0 \\
(b)& \quad 0 \end{align*}$
Work Step by Step
Definitions
$\displaystyle \sinh x=\frac{e^{x}-e^{-x}}{2} \qquad \mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{1}{\sinh x}=\frac{2}{e^{x}-e^{-x}},\quad x\neq 0$
$\displaystyle \cosh x=\frac{e^{x}+e^{-x}}{2} \qquad \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{1}{\cosh x}=\frac{2}{e^{x}+e^{-x}}$
$\displaystyle \tanh x=\frac{\sinh x}{\cosh x} =\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \qquad \coth x=\frac{1}{\tanh x}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}},\quad x\neq 0$
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(a)
$\displaystyle \sinh^{-1}0=\frac{e^{0}-e^{0}}{2}=0$
(b)
$\displaystyle \tanh^{-1}0=\frac{\sinh x}{\cosh x}=\frac{0}{e^{0}+e^{0}}=0$
