Prove that for all real numbers $x$ and any non-negative integer $n,$ we have that $$(1-\cos x) \, |\sin(x)+\sin(2x) +\dotsc +\sin(nx)| \cdot | \cos(x)+\cos(2x)+\dotsc+\cos(nx)| \le 2.$$
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Please provide context. Where did you encounter the problem? What are your thoughts on the problem? What have you tried? – Ben Grossmann Jun 30 '20 at 14:22
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1Use identity for sum sines and cosines in A.P. – UmbQbify Jun 30 '20 at 14:23
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1Either use complex numbers or use Lagrange's identities. – Ben Grossmann Jun 30 '20 at 14:25
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Note also that $$ 1 - \cos x = 2 \sin^2(x/2) $$ – Ben Grossmann Jun 30 '20 at 14:27
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@Omnomnomnom, any source regarding use of complex numbers(in multiplication series)? – UmbQbify Jun 30 '20 at 14:35
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@user675453 see this post for instance – Ben Grossmann Jun 30 '20 at 14:40
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Now I got : $|\sin^2 \frac{nx}{2} \cdot \sin(nx+x) | \le 2. $ – mmathh uuserr Jun 30 '20 at 14:40
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1@mmathh uuserr, yup you got it. Product of two sine function can be atmost 1 – UmbQbify Jun 30 '20 at 14:42