I think a fun example of this is the simplification of $$\sum_{k=1}^n \sin kx.$$ First multiply by $\sin \frac x2$ to get $$\sum_{k=1}^n \sin kx \sin \frac x2 = \sum_{k=1}^n \frac 12 \left[ \cos (k - \frac 12)x - \cos(k + \frac 12)x \right] = \frac{\cos \frac x2 - \cos(n + \frac 12)x}2$$ then divide by the same term to find $$\sum_{k=1}^n \sin kx = \frac{\cos \frac x2 - \cos(n + \frac 12)x}{2\sin \frac x2}.$$