Evaluate: $\frac{1+tan^{2} A}{1+cot^{2} A}$

Solution:

Evaluate $\frac{1+tan^{2} A}{1+cot^{2} A}$

$ \begin{array}{l} \frac{1+tan^{2} A}{1+cot^{2} A}\ \ =\ \frac{sec^{2} A}{cosec^{2} A}\ \ =\frac{\frac{1}{cos^{2} A}}{\frac{1}{sin^{2} A}}\ \ =\frac{sin^{2} A}{cos^{2} A}\ \ =tan^{2} A \end{array}$