1. **State the problem:** Simplify the expression $$\frac{1+\tan^2(x)}{1+\cot^2(x)}$$. 2. **Recall the Pythagorean identities:** - $$1 + \tan^2(x) = \sec^2(x)$$ - $$1 + \cot^2(x) = \csc^2(x)$$ 3. **Substitute these identities into the expression:** $$\frac{1+\tan^2(x)}{1+\cot^2(x)} = \frac{\sec^2(x)}{\csc^2(x)}$$ 4. **Rewrite secant and cosecant in terms of sine and cosine:** $$\sec^2(x) = \frac{1}{\cos^2(x)}, \quad \csc^2(x) = \frac{1}{\sin^2(x)}$$ 5. **Substitute these into the fraction:** $$\frac{\frac{1}{\cos^2(x)}}{\frac{1}{\sin^2(x)}}$$ 6. **Divide the fractions:** $$= \frac{1}{\cos^2(x)} \times \frac{\sin^2(x)}{1} = \frac{\sin^2(x)}{\cos^2(x)}$$ 7. **Recognize the result:** $$\frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x)$$ **Final answer:** $$\boxed{\tan^2(x)}$$