$\dfrac{\cot A + \tan B}{\cot B + \tan A} = ?$

Writing everything in sine and cosine, $\dfrac{\cot A+\tan B}{\cot B+\tan A} = \dfrac{\frac{\cos A}{\sin A}+\frac{\sin B}{\cos B}}{\frac{\cos B}{\sin B}+\frac{\sin A}{\cos A}} = \dfrac{\cos(A-B)/(\sin A\cos B)}{\cos(A-B)/(\sin B\cos A)} = \dfrac{\sin B\cos A}{\sin A\cos B} = \cot A\cdot\tan B$.