query: Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$
positive_passages.docid: 17769
positive_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: 21941
negative_passages.text: Proving: $x = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$
negative_passages.docid: 123920
negative_passages.text: How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$
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: 12741
negative_passages.text: How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$
negative_passages.docid: 104384
negative_passages.text: Trigonometric identity: $\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$
negative_passages.docid: 150581
negative_passages.text: How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$
negative_passages.docid: 13297
negative_passages.text: Prove the following trigonometric identity without a calculator involved
negative_passages.docid: 51979
negative_passages.text: Is there a geometrical method to prove $x<\frac{\sin x +\tan x}{2}$?
negative_passages.docid: 147641
negative_passages.text: How to evaluate $\int \frac { \sin x+\cos x }{ \sin^4 x+\cos^4x}\, dx$?
negative_passages.docid: 98745
negative_passages.text: To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$
negative_passages.docid: 79119
negative_passages.text: Prove that: $1-\frac 12+\frac 13-\frac 14+...+\frac {1}{199}- \frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}$
negative_passages.docid: 52413
negative_passages.text: Show that $\tan{(\pi/7)} \tan{(2\pi/7)}\tan{(3\pi/7)}=\sqrt{7}$
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: 116580
negative_passages.text: Prove by induction: $\sin{x}+\sin{3x}+\dots+\sin{(2n-1)x}=\frac{\sin^2{nx}}{\sin{x}}$
negative_passages.docid: 105238
negative_passages.text: Proving $\tan(\theta_1+\theta_2+\cdots+\theta_n)$ has the form $\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$
negative_passages.docid: 24656
negative_passages.text: Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$