query: Prove $|\sin x - \sin y|< |x-y|$
positive_passages.docid: 143072
positive_passages.text: Proving $|\sin x - \sin y| < |x - y|$
negative_passages.docid: 124208
negative_passages.text: Proof $|\sin(x) - x| \le \frac{1}{3.2}|x|^3$
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 122944
negative_passages.text: Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
negative_passages.docid: 75888
negative_passages.text: How to prove the function $y = \sin x$ is not a closed function?
negative_passages.docid: 118687
negative_passages.text: Prove an inequality with a $\sin$ function: $\sin(x) > \frac2\pi x$ for $0<x<\frac\pi2$
negative_passages.docid: 9986
negative_passages.text: Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
negative_passages.docid: 152495
negative_passages.text: Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
negative_passages.docid: 127759
negative_passages.text: Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
negative_passages.docid: 127380
negative_passages.text: Use vectors to prove $\cos(\beta - \alpha) = \cos \alpha \cos\beta + \sin\alpha \sin\beta$
negative_passages.docid: 120148
negative_passages.text: prove this properties of triangles trigonometric question
negative_passages.docid: 80009
negative_passages.text: Prove that $| |a|-|b| | \le |a-b|$
negative_passages.docid: 16253
negative_passages.text: Proof that $|\sin(nx)| \le n|\sin(x)|$
negative_passages.docid: 24656
negative_passages.text: Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
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: 150819
negative_passages.text: How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$