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

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