Where did cotQ come from in canonical transformation?

[M. next]

I won't write the whole thing(unless asked for from you guys), but I just want to ask where did cotQ come from in canonical transformation.

For instance, F(q,Q) = 1/2p$_{(q,Q)}$q => $\partial$F/$\partial$q=1/2p+1/2*$\partial$p/$\partial$q=1/2(p+qcotQ)=p.
What is this? How did the last things come from??

 

[vanhees71]

How should we know, if you don't give the generator of the transformation explicitly?

 

[M. next]

Sorry my bad, I didn't know why this was important (if u may also clear that) but it is F=1/2*p*q, and this was at the end of an exercise, and what I posted in my original question, was a remark under the question! But I guess related to it

 

[M. next]

I am struggling alone in understanding classical mechanics!