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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 ?

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How about using $$ 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right) $$ ? – Sangchul Lee Nov 24 '12 at 17:33
up vote 2 down vote accepted

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 $2$.

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Thank you! What a pity I didn't notice that... – Chon Nov 25 '12 at 20:13

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