query: Why is it valid to say $\frac{\sin x}x$ is the product of the linear factors given by its roots?
positive_passages.docid: 24311
positive_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: 13747
negative_passages.text: Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
negative_passages.docid: 146594
negative_passages.text: Prove that $\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}dx$
negative_passages.docid: 149447
negative_passages.text: Proving $\sin (x)=\cos (90^\circ-x)$
negative_passages.docid: 78101
negative_passages.text: Does $\sum\frac{\sin n}{n}$ converge?
negative_passages.docid: 123476
negative_passages.text: Prove $[\sin x]' = \cos x$ without using $\lim\limits_{x\to 0}\frac{\sin x}{x} = 1$
negative_passages.docid: 34650
negative_passages.text: Explain why $\int_0^\infty\frac{\sin{4x}}{x}\prod\limits_{k=1}^n \cos\left(\frac{x}{k}\right) dx\approx\frac{\pi}{2}$
negative_passages.docid: 53307
negative_passages.text: how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
negative_passages.docid: 114418
negative_passages.text: A deduction from $x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$
negative_passages.docid: 83851
negative_passages.text: Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?
negative_passages.docid: 64941
negative_passages.text: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
negative_passages.docid: 19561
negative_passages.text: Why is $\frac{1}{\frac{1}{X}}=X$?
negative_passages.docid: 74885
negative_passages.text: How is Leibniz's rule for the derivative of a product related to the binomial formula?
negative_passages.docid: 71156
negative_passages.text: $f(x) = \frac {ax+b}{cx+d}$ , bijection $f: \Bbb R \to\Bbb R$?
negative_passages.docid: 111102
negative_passages.text: Prove the Trigonometric Identity: $\frac{\sin x}{1-\cos x} = \frac{1+\cos x}{\sin x}$
negative_passages.docid: 56805
negative_passages.text: Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of "the functions one finds on a calculator"?
negative_passages.docid: 123392
negative_passages.text: How does $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ make sense?