Answer
The curvature at the point $(0,1,0) $ is:
$$
\kappa (0)=\frac{1}{2 \cosh ^{2} (0)}=\frac{1}{2},
$$
The torsion at the point $(0,1,0) $ is:
$$
\tau (0)=\frac{-1}{2 \cosh ^{2} t}=\frac{-1}{2}.
$$
Work Step by Step
The parametric equations of the curve are:
$$
x=\sinh t, \ \ y=\cosh t, \ \ z=t
$$
So, the curve is:
$$
\mathrm{r}=\left\langle \sinh t, \cosh t, t \right\rangle
$$
The curvature formula of the curve is given by:
$$\kappa=\left|\frac{d \mathbf{T}}{d s}\right|=\frac{\left|\mathbf{T}^{\prime}(t)\right|}{\left|\mathbf{r}^{\prime}(t)\right|}=\frac{\left|\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right|}{\left|\mathbf{r}^{\prime}(t)\right|^{3}}
$$
We first compute the ingredients needed
$$
\begin{aligned}
\mathrm{r}^{\prime} &=\left\langle \cosh t, \sinh t, 1 \right\rangle,\\
\mathrm{r}^{\prime \prime}&=\langle \sinh t,\cosh t, 0 \rangle,\\
\mathrm{r}^{\prime \prime \prime}&=\langle \cosh t ,\sinh t ,0\rangle
\end{aligned}
$$
and
$$
\begin{aligned}
\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime} &=\left\langle-\cosh t, \sinh t, \cosh ^{2} t-\sinh ^{2} t\right\rangle \\
&=\langle-\cosh t, \sinh t, 1\rangle
\end{aligned}
$$
The curvature formula of the given curve is:
$$
\begin{aligned}
\kappa &=\frac{\left|\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right|}{\left|\mathbf{r}^{\prime}\right|^{3}} \\
&=\frac{|(-\cosh t, \sinh t, 1\rangle|}{|\langle\cosh t, \sinh t, 1)|^{3}} \\
&=\frac{\sqrt{\cosh ^{2} t+\sinh ^{2} t+1}}{\left(\cosh ^{2} t+\sinh ^{2} t+1\right)^{3 / 2}} \\
&=\frac{1}{\cosh ^{2} t+\sinh ^{2} t+1} \\
&=\frac{1}{2 \cosh ^{2} t}.
\end{aligned}
$$
The torsion of the given curve, by using the formula in Exercise 63(d), is:
$$
\begin{aligned}
\tau &=\frac{\left(\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right) \cdot \mathbf{r}^{\prime \prime \prime}}{\left|\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right|^{2}} \\
&=\frac{\langle-\cosh t, \sinh t, 1\rangle \cdot\langle\cosh t, \sinh t, 0\rangle}{\cosh ^{2} t+\sinh ^{2} t+1} \\
&=\frac{-\cosh ^{2} t+\sinh ^{2} t}{2 \cosh ^{2} t} \\
&=\frac{-1}{2 \cosh ^{2} t}
\end{aligned}
$$
Thus, the curvature at the point $(0,1,0) $ is:
$$
\kappa (0)=\frac{1}{2 \cosh ^{2} (0)}=\frac{1}{2},
$$
The torsion at the point $(0,1,0) $ is:
$$
\tau (0)=\frac{-1}{2 \cosh ^{2} t}=\frac{-1}{2}.
$$
