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I would like to calculate:

$$ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x°)}\right) $$

Here is what I found:

$$ 1-\frac{1}{\tan(x)}=\frac{\sqrt{2}\sin(x-45°)}{\sin(x)} $$

$$ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x°)}\right)=\prod_{x=1}^{44} \frac{\sqrt{2}\sin(x-45°)}{\sin(x)}=2^{22} \frac{\prod_{x=1}^{44} \sin(x-45°)}{\prod_{x=1}^{44} \sin(x)} $$

$$ x\rightarrow 45°-x $$

$$ \prod_{x=1}^{44}\left(1-\frac{1}{\tan(x°)}\right)=\frac{\prod_{x=1}^{44} -\sin(x)}{\prod_{x=1}^{44} \sin(x)}=2^{22} $$

  • 6
    So what is the question? You seem to have your answer.2011-12-29
  • 0
    That looks right to me, and my computer confirms it. Nice one, that's quite an elegant result!2011-12-29
  • 3
    This is a variant of http://math.stackexchange.com/questions/75825/a-fast-way-for-computing-prod-limits-i-1451-tan-i-circ/75834#75834 (perhaps even the same..).2011-12-29

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