query: Prove that $-x\sin x<\frac{x^4}{6}-x^2$ for all $x\in(0,\pi)$
positive_passages.docid: 89265
positive_passages.text: Proving that $x - \frac{x^3}{3!} 0$
negative_passages.docid: 60559
negative_passages.text: How prove this $x^3<\sin^2{x}\tan{x},x\in\left(0,\dfrac{\pi}{2}\right)$
negative_passages.docid: 9986
negative_passages.text: Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
negative_passages.docid: 11387
negative_passages.text: Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$
negative_passages.docid: 78599
negative_passages.text: How to prove that $\frac{b-a}{\sqrt{1-a^2}}<\sin^{-1}b-\sin^{-1}a<\frac{b-a}{\sqrt{1-b^2}}$ by using Mean Value Theorem?
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: 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: 45619
negative_passages.text: Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$
negative_passages.docid: 26301
negative_passages.text: Prove that $x^3 \equiv x \bmod 6$ for all integers $x$
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: 104688
negative_passages.text: How can I prove that $x-{x^2\over2}<\ln(1+x)$
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 120795
negative_passages.text: Showing that $\sin^2x\cdot\sin^22x\cdot\sin^24x\cdot\sin^28x\cdots\sin^22^nx\leq\frac{3^n}{4^n}$
negative_passages.docid: 130078
negative_passages.text: How can one prove that $\pi^4 + \pi^5 < e^6$?
negative_passages.docid: 50509
negative_passages.text: Prove that $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}}~dx=0$ (without trigonometric substitution)
negative_passages.docid: 32823
negative_passages.text: Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$
negative_passages.docid: 143072
negative_passages.text: Proving $|\sin x - \sin y| < |x - y|$