Verify sinx-siny/cosx-cosy=-cot ((x+y) / 2)

Sin (A+B) = sinAcosB+cosAsinB

sin (A-B) = sinAcosB-cosAsinB

sin (A+B) - sin (A-B) = 2cosAsinB

So, if x=A+B and y=A-B, A = (x+y) / 2, B = (x-y) / 2

sinx-siny=2cosAsinB=2cos ((x+y) / 2) sin ((x-y) / 2)

cos (A+B) = cosAcosB-sinAsinB

cos (A-B) = cosAcosB+sinAsinB

cos (A+B) - cos (A-B) = - 2sinAsinB

cosx-cosy=-2sinAsinB=-2sin ((x+y) / 2) sin ((x-y) / 2).

So, (sinx-siny) / (cosx-cosy) = - cot ((x+y) / 2) QED