A Curious Identity

Asked Feb 13 '14 at 23:10

Active Feb 14 '14 at 22:45

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I met the following equations when I was trying to solve a complex line integral (W.Rudin, RCA, p.228 ex.13). My question is how to prove them:

We have to show that for $n>2$ even

$$ 2^{n/2}\prod^{\left(n - 2\right)/2}_{k = 1} \left[1 - \cos\left(2k\pi \over n\right)\right] = n, $$

and for $n>1$ odd,

$$ 2^{\left(n - 1\right)/2}\prod_{k = 1}^{\left(n - 1\right)/2} \left[1 - \cos\left(2k\pi \over n\right)\right] =n\text{ ?}$$

-This can be rewritten as a curious identity after the below hint of Greg Martin-

Thx.

edited Feb 14 '14 at 22:45

Felix Marin

asked Feb 13 '14 at 23:10

btm

1

Hint: $2(1-\cos x) = 2-e^{ix}-e^{-ix} = (1-e^{ix})(1-e^{-ix})$. – Greg Martin Feb 13 '14 at 23:51
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http://math.stackexchange.com/questions/70231/how-to-prove-those-curious-identities – btm Feb 14 '14 at 21:27
For an elementary method, take a look here – chubakueno Feb 14 '14 at 22:12

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