Show that sin S - sin T = 2.cos (S + T / 2) .sin (S - T / 2)
(By the way, am I right in saying that you are not allowed to expand the RHS when the question asks to 'show'?)
Thanks !
On the right side
$\displaystyle \cos \left(\frac{S}{2}+\frac{T}{2}\right) = \cos \frac{S}{2} \cos \frac{T}{2} - \sin \frac{S}{2} \sin \frac{T}{2}$
$\displaystyle \sin \left(\frac{S}{2}-\frac{T}{2}\right) = \sin \frac{S}{2} \cos \frac{T}{2} - \cos \frac{S}{2} \sin \frac{T}{2}$
Multiply both to get the RHS
On the left side
$\displaystyle \sin S - \sin T = 2 \sin \frac{S}{2} \cos \frac{S}{2} - 2 \sin \frac{T}{2} \cos \frac{T}{2}$