Answer
The solutions are $x=\left\{-1, 1\right\}$.
Work Step by Step
Use the property $a^m = a^n \longrightarrow m=n$ to obtain:
$\begin{array}{ccc}
&x^2-1&=&1-x^2
\\&x^2-1+x^2&=&1-x^2+x^2
\\&2x^2-1&=&1
\\&2x^2-1+1&=&1+1
\\&2x^2&=&2
\\&\dfrac{2x^2}{2} &= &\dfrac{2}{2}
\\&x^2&=&1
\\&\sqrt{x^2}&=&\pm\sqrt{1}
\\&x&=&\pm1
\end{array}$
Thus, the solutions are $x=\left\{-1, 1\right\}$.
