Question

Show that cot((x+y)/2) = - (sin x - sin y)/(cos x - cos y) for all values of x and y for which both sides are defined.

I tried manipulating both sides in terms of trig identities but I don't really have a solution....help would be appreciated, thanks.

Answer 1

Using the sum-to-product formulas,

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

now just divide to get

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

Answer 2

Two of the standard conversion formulas are:

sinA - sinB = 2sin((A-B)/2) cos( (A+B)/2)
and
cosA - cosB = - 2sin( (A-B)/2) sin( (A+B)/2)

see:
near bottom of page under:
Sum-to-Product Formulas
http://www.sosmath.com/trig/Trig5/trig5/trig5.html

RS
= -(sinx - siny)/(cosx - cosy)
= - 2sin((X-Y)/2) cos( (X+Y)/2)/- 2sin( (X-Y)/2) sin( (X+Y)/2)
= cos((X+Y)/2) / sin(X+Y)/2)
= cot((X+Y)/2)

= LS