I'm trying to show that:
$ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$
It is equivalent to show that:
$ (1-\cos x) \left (\frac{\sin \left(\frac{nx}{2} \right)}{ \sin \left( \frac{x}{2} \right)} \right)^2 |\sin((n+1)x)|\leq 4 $
Any idea ?