1. **State the problem:** Simplify the expression $ \frac{x^2 - 1}{x^2 - 1} = 1 $. 2. **Recall the formula and rules:** When the numerator and denominator of a fraction are the same (and not zero), the fraction equals 1. 3. **Factor the numerator and denominator:** $$x^2 - 1 = (x + 1)(x - 1)$$ 4. **Rewrite the fraction:** $$\frac{(x + 1)(x - 1)}{(x + 1)(x - 1)}$$ 5. **Cancel common factors:** $$\frac{\cancel{(x + 1)}\cancel{(x - 1)}}{\cancel{(x + 1)}\cancel{(x - 1)}} = 1$$ 6. **Important note:** The expression is valid only if $x \neq 1$ and $x \neq -1$ because the denominator cannot be zero. 7. **Final answer:** $$\frac{x^2 - 1}{x^2 - 1} = 1 \quad \text{for} \quad x \neq \pm 1$$