query: If $\alpha + \beta + \gamma = \pi$ prove that $\sin(2\alpha)+\sin(2\beta) + \sin(2\gamma) = 4\sin(\alpha)\sin(\beta)\sin(\gamma)$
positive_passages.docid: 59014
positive_passages.text: Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
negative_passages.docid: 8919
negative_passages.text: If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha.$
negative_passages.docid: 142168
negative_passages.text: can we a prove ⊢ (α → α) → (α → α)
negative_passages.docid: 61343
negative_passages.text: Prove that if $\alpha, \beta, \gamma$ are angles in triangle, then $(tan(\frac{\alpha}{2}))^2+(tan(\frac{\beta}{2}))^2+(tan(\frac{\gamma}{2}))^2\geq1$
negative_passages.docid: 147684
negative_passages.text: Prove that $\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$ using given condition
negative_passages.docid: 50777
negative_passages.text: Prove that $4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$ .
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 69082
negative_passages.text: Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$
negative_passages.docid: 145632
negative_passages.text: Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$
negative_passages.docid: 148744
negative_passages.text: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
negative_passages.docid: 44119
negative_passages.text: Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
negative_passages.docid: 51979
negative_passages.text: Is there a geometrical method to prove $x<\frac{\sin x +\tan x}{2}$?
negative_passages.docid: 129880
negative_passages.text: Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
negative_passages.docid: 70765
negative_passages.text: If $\frac{\tan(a+b-c)}{\tan(a-b+c)}=\frac{\tan c}{\tan b}$, then $\sin(b-c)=0$ or $\sin 2a + \sin2b + \sin 2c=0$
negative_passages.docid: 47356
negative_passages.text: If : $\tan^2\alpha \tan^2\beta +\tan^2\beta \tan^2\gamma + \tan^2\gamma \tan^2\alpha + 2\tan^2\alpha \tan^2\beta \tan^2\gamma =1\dots$
negative_passages.docid: 43219
negative_passages.text: If $[K(\alpha):K]=p\neq q=[K(\beta):K]$ then $[K(\alpha+\beta):K]=pq$
negative_passages.docid: 112969
negative_passages.text: Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$