How do you verify the identity #(sinx+siny)/(cosx-cosy)=-cot((x-y)/2)#?

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Answer 1 · 2017-01-06T16:56:55

Please see below.

Using identities #sinA+sinB=2sin((A+B)/2)cos((A-B)/2)# and #cosA-cosB=2sin((A+B)/2)sin((B-A)/2)#, hence

#(sinx+siny)/(cosx-cosy)#

= #(2sin((x+y)/2)cos((x-y)/2))/(2sin((x+y)/2)sin((y-x)/2))#

= #(cos((x-y)/2))/(sin((y-x)/2))#

and as #sin(-A)=-sinA#, we have #sin((y-x)/2)=-sin((x-y)/2)#

Hence above becomes

#(cos((x-y)/2))/(-sin((x-y)/2))=-cot((x-y)/2)#