1. **State the problem:** Prove or simplify the expression $$\sqrt{\frac{1 + \tan^2 A}{1 + \cot^2 A}} = \tan A$$. 2. **Recall the Pythagorean identities:** - $$1 + \tan^2 A = \sec^2 A$$ - $$1 + \cot^2 A = \csc^2 A$$ 3. **Substitute these identities into the expression:** $$\sqrt{\frac{\sec^2 A}{\csc^2 A}}$$ 4. **Rewrite the square root of a fraction as the fraction of square roots:** $$\frac{\sqrt{\sec^2 A}}{\sqrt{\csc^2 A}}$$ 5. **Simplify the square roots:** $$\frac{\sec A}{\csc A}$$ 6. **Express secant and cosecant in terms of sine and cosine:** $$\frac{\frac{1}{\cos A}}{\frac{1}{\sin A}} = \frac{1}{\cos A} \times \frac{\sin A}{1} = \frac{\sin A}{\cos A}$$ 7. **Recognize that $$\frac{\sin A}{\cos A} = \tan A$$, so the original expression simplifies to $$\tan A$$. **Final answer:** $$\sqrt{\frac{1 + \tan^2 A}{1 + \cot^2 A}} = \tan A$$