query: Prove $\cos(\sin x)>\sin(\cos x)$
positive_passages.docid: 2648
positive_passages.text: Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.
negative_passages.docid: 100777
negative_passages.text: Finding the derivatives of sin(x) and cos(x)
negative_passages.docid: 156881
negative_passages.text: How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$
negative_passages.docid: 40235
negative_passages.text: How would I differentiate $\sin{x}^{\cos{x}}?$
negative_passages.docid: 132448
negative_passages.text: Prove that: $\int_0^{n\pi}f(\cos^2(x))\,dx=n\int_0^{\pi}f(\cos^2(x))\,dx?$
negative_passages.docid: 84730
negative_passages.text: use the fact that $\tan 2x=\sin2x \ /\cos2x$ to prove that $\tan 2x=2\tan x/(1-\tan^2x)$
negative_passages.docid: 96388
negative_passages.text: Prove $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$
negative_passages.docid: 9986
negative_passages.text: Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $
negative_passages.docid: 37040
negative_passages.text: Prove reduction formula for $\int \cos^n (x)\sin^m (x) \, dx$
negative_passages.docid: 104321
negative_passages.text: Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$
negative_passages.docid: 17769
negative_passages.text: Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$
negative_passages.docid: 46353
negative_passages.text: Proof that $\cos(\pi/3)=1/2$
negative_passages.docid: 25569
negative_passages.text: Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
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: 120444
negative_passages.text: Trigonometric Triangle Equality
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: 42545
negative_passages.text: If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$