query: Prove $\sin(2\pi/7) +\sin(4\pi/7) +\sin(8\pi/7) = \sqrt7/2$
positive_passages.docid: 17168
positive_passages.text: Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$
negative_passages.docid: 75679
negative_passages.text: How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$
negative_passages.docid: 131310
negative_passages.text: Evaluating $\dfrac{1}{\sin(2x)} + \dfrac{1}{\sin(4x)} + \dfrac{1}{\sin(8x)} + \dfrac{1}{\sin(16x)}$
negative_passages.docid: 95085
negative_passages.text: Prove by induction $ \sin x + \sin 2x + ... + \sin nx = \frac {\sin (\frac {n + 1} {2} x)} {\sin \frac{x}{2}} \sin \frac{nx}{2} $
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: 111295
negative_passages.text: Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
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: 26474
negative_passages.text: how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$?
negative_passages.docid: 100895
negative_passages.text: Prove $1 + \cos 2C - \cos 2A - \cos 2B = 4\sin A\sin B\cos C$ for $A$, $B$, $C$ the angles of a triangle
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: 19216
negative_passages.text: Proving that $\arctan(x)+\arctan(1/x)=\pm \pi/2$, could this line of reasoning possibly be correct?
negative_passages.docid: 1859
negative_passages.text: Proving $\arctan(\sqrt{x^2+1}-x)=\frac{\pi}{4}-\frac{1}{2}\arctan(x)$
negative_passages.docid: 81750
negative_passages.text: $4 \sin 72^\circ \sin 36^\circ = \sqrt 5$
negative_passages.docid: 45227
negative_passages.text: If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt.
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: 85944
negative_passages.text: Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $
negative_passages.docid: 53307
negative_passages.text: how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$