query: Product Identity Multiple Angle or $\sin(nx)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(\frac{k\pi}n+x\right)$
positive_passages.docid: 137844
positive_passages.text: Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$
negative_passages.docid: 23919
negative_passages.text: Trying to evaluate $\prod_{k=1}^{n-1}(1-e^{2k\pi i/n})$ for my complex analysis homework
negative_passages.docid: 60098
negative_passages.text: How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?
negative_passages.docid: 51679
negative_passages.text: Simplifying the product $\prod\limits_{k=2}^n \left(1-\frac1{k^2}\right)$
negative_passages.docid: 146035
negative_passages.text: Proving $\sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =\sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0. $
negative_passages.docid: 83080
negative_passages.text: Question regarding $f(n)=\cot^2\left(\frac\pi n\right)+\cot^2\left(\frac{2\pi}n\right)+\cdots+\cot^2\left(\frac{(n-1)\pi}n\right)$
negative_passages.docid: 79685
negative_passages.text: For natural $n$, prove $\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}$
negative_passages.docid: 107067
negative_passages.text: Finding the argument of $z=1-\cos(2\theta)-i\sin(2\theta)$
negative_passages.docid: 49233
negative_passages.text: Find $cot\space x_n$ if $sin(x_{n+1}-x_n)+2^{-(n+1)}sin\space x_nsin \space x_{n+1}=0$ for $n\ge1$
negative_passages.docid: 133880
negative_passages.text: $|\sin^2x-\sin^2y|\le K|x-y|$ find K
negative_passages.docid: 147886
negative_passages.text: Proof of the angle sum identity for $\sin$
negative_passages.docid: 24311
negative_passages.text: Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
negative_passages.docid: 281
negative_passages.text: How one can prove $\prod_{k=1}^{n-1} (\cos\theta-\cos(k\pi/n)) = 2^{1-n}\sin(n\theta)\csc\theta$?
negative_passages.docid: 104427
negative_passages.text: Sines and cosines of angles in arithmetic progression
negative_passages.docid: 14386
negative_passages.text: Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$
negative_passages.docid: 141787
negative_passages.text: Anybody know a proof of $\prod_{n=1}^\infty\cos(x/2^n)=\sin x/x$.
negative_passages.docid: 42545
negative_passages.text: If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$