How do I verify -2cot ((x-y) /2) = (2sin(x) + 2sin(y)) / (cos(x) - cos(y)) by only simplifying one side of the equation?

Answer 1

We can start with the right side of the equation,

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

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

(-2(sin((x+y)/2))(sin((x-y)/2)))

(above is making use of sum-to-product formula)

reducing same terms will give the below

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

=-2cot((x-y)/2)