cot(x+y)=cos(x+y)/sin(x+y)=(cos(x)cos(y)-sin(x)sin(y))/(sin(x)cos(y)+cos(x)sin(y)).
This can be taken further:
Divide through by sin(x):
(cot(x)cos(y)-sin(y))/(cos(y)+cot(x)sin(y)) then by sin(y):
(cot(x)cot(y)-1)/(cot(y)+cot(x)). This is an alternative expression for cot(x+y).
We can take it further again by multiplying through by tan(x)tan(y):
(1-tan(x)tan(y))/(tan(x)+tan(y)).