1. **State the problem:** Evaluate the expression $\tan 1^2 - 1$. 2. **Clarify the expression:** The expression can be interpreted as $\tan(1^2) - 1$, which means take the tangent of $1^2$ and then subtract 1. 3. **Calculate the exponent:** $1^2 = 1$. 4. **Apply the tangent function:** $\tan(1)$, where 1 is in radians. 5. **Evaluate $\tan(1)$:** Using a calculator or approximation, $\tan(1) \approx 1.5574$. 6. **Subtract 1:** $1.5574 - 1 = 0.5574$. 7. **Final answer:** $\tan 1^2 - 1 = 0.5574$ (approximately).