If tan A + tan B = a and cot A + cot B = b, then 1/a - 1/b is equal to ____.

tan A + tan B = a
cot A + cot B = b

​$$\frac{1}{\tan A + \tan B } = \frac{1}{a}$$ 

​$$\frac{1}{\cot A + \cot B } = \frac{1}{b}$$​​

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

​$$\frac{1}{\cot A + \cot B } = \frac{1}{b}$$ 

can be written as 

​$$\frac{1}{\frac{1}{\tan A} + \frac{1}{\tan B} } = \frac{1}{b} \rightarrow \frac{1}{b} = \frac{ \tan A \tan B}{ \tan A + \tan B }$$  

Now,  

​$$\frac{1}{a} - \frac{1}{b} = \frac{ 1}{\tan A + \tan B} - \frac{\tan A \tan B}{\tan A + \tan B} \\ \ \\ \ \frac{1}{a} - \frac{1}{b} = \frac{ 1 - \tan A \tan B }{ \tan A + \tan B} \implies \frac{1}{\tan (A + B)} = \cot (A + B)$$ ​