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

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