Showing that $ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$ 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 ?

Using the identity
$$ 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right), $$
we readily identify the left-hand side as
$$ 2 \sin^2 \left(\frac{nx}{2} \right) \left|\sin((n+1)x)\right|, $$
which is clearly less than or equal to $2$.
