Given that

\( \Large \tan A - \tan B = x \) ...(i)

and \( \Large \cot B - \cot A = y \) ...(ii)

Now, \( \Large \cot \left(A-B\right) = \frac{1}{\tan \left(A-B\right) } \)

= \( \Large \frac{1+\tan A \tan B}{\tan A - \tan B} \)

= \( \Large \frac{1}{\tan A - \tan B} + \frac{\tan A \tan B}{\tan A - \tan B} \)

= \( \Large \frac{1}{x}+\frac{1}{y} \) [from I and II]