derive a formula for cot (x+y)

Question

Given: sin(x+y) = sinxcosy+cosxsiny And cos(x+y) = cosxcosy-sinxsiny

Answer 1

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)).

Answer 2

Full solution shown below: