query: :Duplicate: How to prove that $\cos (\pi * z) = \prod_{n \in \mathbb{Z}_{\text{odd}}} (1-\frac{2 z}{n})e^{2z/n}$
positive_passages.docid: 23727
positive_passages.text: How can I deduce $\cos\pi z=\prod_{n=0}^{\infty}(1-4z^2/(2n+1)^2)$?
negative_passages.docid: 114796
negative_passages.text: Direct proof that the product of odd integers is odd
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
negative_passages.docid: 48586
negative_passages.text: Prove: $\cot x=\sin x\,\sin\left(\frac{\pi}{2}-x\right)+\cos^2x\,\cot x$
negative_passages.docid: 12653
negative_passages.text: Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$
negative_passages.docid: 78712
negative_passages.text: Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
negative_passages.docid: 6453
negative_passages.text: Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.
negative_passages.docid: 13864
negative_passages.text: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$
negative_passages.docid: 68858
negative_passages.text: Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$
negative_passages.docid: 107067
negative_passages.text: Finding the argument of $z=1-\cos(2\theta)-i\sin(2\theta)$
negative_passages.docid: 54432
negative_passages.text: What can be said about $\prod_{s=2}^{\infty} \zeta(s) $?
negative_passages.docid: 100320
negative_passages.text: Prove that $\cos\frac {2\pi}{7}+ \cos\frac {4\pi}{7}+ \cos\frac {8\pi}{7}=-\frac{1}{2}$
negative_passages.docid: 56934
negative_passages.text: How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?.
negative_passages.docid: 154544
negative_passages.text: Prove that $ \sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}=\frac{z}{1-z} $
negative_passages.docid: 46353
negative_passages.text: Proof that $\cos(\pi/3)=1/2$
negative_passages.docid: 83280
negative_passages.text: Infinite product $ \prod^{\infty}_{n=0}\left(1-\frac{1}{2^{3n}}-\frac{1}{2^{6n}}\right)$