query: If $\alpha=\beta+\omega$, then $1 + \cos\alpha+\cos\beta+\cos\omega = 4 \cos\frac{\alpha}{2}\cos\frac{\beta}{2}\cos\frac{\omega}{2}$
positive_passages.docid: 11711
positive_passages.text: Prove trigonometry identity for $\cos A+\cos B+\cos C$
negative_passages.docid: 41792
negative_passages.text: If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
negative_passages.docid: 617
negative_passages.text: Find θ: $β=\arctan\frac{(R+r)·\sin(θ)-r·\sin(θ(1+\frac{R}{r}))}{(R+r)·\cos(θ)-r·\cos(θ(1+\frac{R}{r}))}$
negative_passages.docid: 17102
negative_passages.text: If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$
negative_passages.docid: 130542
negative_passages.text: Every natural number $n$ can be written as $n=s-t$ with $\omega(s)=\omega(t)$
negative_passages.docid: 93756
negative_passages.text: $1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$
negative_passages.docid: 43598
negative_passages.text: Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?
negative_passages.docid: 9246
negative_passages.text: Calculating $\prod (\omega^j - \omega^k)$ where $\omega^n=1$.
negative_passages.docid: 37108
negative_passages.text: Solve $\cos(z)=\frac 34+\frac i4$
negative_passages.docid: 23694
negative_passages.text: Given that $abc=1$, prove that $\frac{b}{c} + \frac{c}{a}+ \frac{a}{b} \ge \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$.
negative_passages.docid: 104321
negative_passages.text: Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$
negative_passages.docid: 105117
negative_passages.text: $1 + \frac{1}{3}\frac{1}{4} + \frac{1}{5}\frac{1}{4^2} + \frac{1}{7}\frac{1}{4^3} + ..........$
negative_passages.docid: 68743
negative_passages.text: If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ then $\frac{1}{a^5}+\frac{1}{b^5}+\frac{1}{c^5}=\frac{1}{a^5+b^5+c^5}.$
negative_passages.docid: 100608
negative_passages.text: Prove that $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{1/2}(\partial \Omega )}))$
negative_passages.docid: 9539
negative_passages.text: $\alpha$ and $\beta$ are solution of $a \cdot \tan\theta + b \cdot \sec\theta = c$ show $ \tan(\alpha + \beta) = \frac{2ac}{a^2 - c^2}$
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: 42677
negative_passages.text: Proving $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in a finite field