Factor out constants

Factor 50 from the numerator and 25 from the denominator to simplify the expression:

$\frac{50(1 + \cot^2 A)}{25(1 + \tan^2 A)} = 2 \cdot \frac{1 + \cot^2 A}{1 + \tan^2 A}$

Apply trigonometric identities

Substitute $1 + \cot^2 A = \csc^2 A$ and $1 + \tan^2 A = \sec^2 A$:

$2 \cdot \frac{\csc^2 A}{\sec^2 A} = 2 \cdot \frac{1/\sin^2 A}{1/\cos^2 A} = 2 \cdot \frac{\cos^2 A}{\sin^2 A}$

Final simplification

Since $\frac{\cos^2 A}{\sin^2 A} = \cot^2 A$, the final value is:

$2 \cot^2 A$