1. **State the problem:** Given that $\tan \theta + \cot \theta = 5$, find the value of $\tan^2 \theta + \cot^2 \theta$. 2. **Recall the formula:** We know that $$ (\tan \theta + \cot \theta)^2 = \tan^2 \theta + 2 + \cot^2 \theta $$ This comes from expanding the square: $$(a+b)^2 = a^2 + 2ab + b^2$$ where $a = \tan \theta$ and $b = \cot \theta$. 3. **Use the given value:** Substitute $\tan \theta + \cot \theta = 5$ into the formula: $$ 5^2 = \tan^2 \theta + 2 + \cot^2 \theta $$ which simplifies to $$ 25 = \tan^2 \theta + \cot^2 \theta + 2 $$ 4. **Solve for $\tan^2 \theta + \cot^2 \theta$:** $$ \tan^2 \theta + \cot^2 \theta = 25 - 2 = 23 $$ **Final answer:** $$ \boxed{23} $$