If $\tan A+\tan B=p$ and $\cot A+\cot B=q$ then $\cot \left(A+B\right)=$  

$\cot A+\cot B=q\to \left(2\right)$

$\Rightarrow \frac{1}{\tan A}+\frac{1}{\tan B}=q$

$\Rightarrow \frac{\tan A+\tan B}{\tan A.\tan B}=q$

$\Rightarrow$ $\boxed{\tan A.\tan B=\frac{p}{q}}$

Now $\cot \left(A+B\right)=\frac{1}{\tan \left(A+B\right)}$

$=\frac{1-\tan A.\tan B}{\tan A+\tan B}$

$=\frac{1-\frac{p}{q}}{p}$

$\Rightarrow$ $\boxed{\cot \left(A+B\right)=\frac{q-p}{pq}}$