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July 13th, 2017, 08:13 PM   #1
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Trigonometric Product

If $\displaystyle \theta = \frac{2\pi}{2009},$ Then $\displaystyle \cos \theta \cdot \cos 2\theta\cdot \cos 3 \theta \cdot\cdots \cos 1004 \theta$ is
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July 16th, 2017, 12:51 PM   #2
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1/2^1004
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July 16th, 2017, 01:49 PM   #3
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This can be easily solved using complex numbers, let me start by giving a hint:

$$z^n - 1 = \prod_{\nu = 1}^n (z - e^{2\pi\nu/n})$$

Try to rewrite the right hand side to the product of cosines.
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July 17th, 2017, 03:10 AM   #4
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If $\theta = 2\pi/9$, using cos(A) ≡ sin(2A)/(2sin(A)) for non-zero A,
$\cos(\theta)\cos(2\theta) \cos(3\theta)\cos(4\theta) = \dfrac{\sin(4\pi/9)\sin(8\pi/9)\sin(12\pi/9)\sin(16\pi/9)}{16\sin(2\pi/9)\sin(4\pi/9)\sin(6\pi/9)\sin(8\pi/9)} = 1/16$.

The same method applies to the original problem.
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