Answer
$\frac{x-1}{x^2-1}=1$
Work Step by Step
The rational equation $\frac{x-1}{x^2-1}=1$ has one solution and one extraneous solution as seen from the solution below.
By cross-multiplication, the rational equation $\frac{x-1}{x^2-1}=1$ is equivalent to
$$
x-1=x^2-1\tag{1}
.$$
Solving for $x$ results in
$$\begin{aligned}
x^2-x&=0
\\
x(x-1)&=0
\end{aligned}
\\
\begin{array}{l|r}
x=0, & x-1=0
\\
& x=1.
\end{array}$$
Checking:
$$\begin{aligned}
\text{When }x=0&:
\\
\frac{0-1}{0^2-1}&\overset{?}=1
\\
\frac{-1}{-1}&\overset{?}=1
\\
1&\overset{\checkmark}=1
\\\\
\text{When }x=1&:
\\
\frac{1-1}{1^2-1}&\overset{?}=1
\\
\frac{0}{0}&\overset{?}=1&\text{(undefined)}
.\end{aligned}$$
Hence, $x=0$ is a solution while $x=1$ is an extraneous solution.
