$\dfrac{\sec A + \tan A}{\operatorname{cosec} A + \cot A} = ?$

Multiplying numerator and denominator by the respective conjugates and using $\sec^2 A - \tan^2 A = 1$ and $\operatorname{cosec}^2 A - \cot^2 A = 1$, the ratio rearranges to $\dfrac{\operatorname{cosec} A - \cot A}{\sec A - \tan A}$.