query: Does $-\frac12\cos(2x)=\sin^2(x)$?
positive_passages.docid: 29370
positive_passages.text: Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) \, dx $
negative_passages.docid: 82321
negative_passages.text: Is $\cos(x) \geq 1 - \frac{x^2}{2}$ for all $x$ in $R$?
negative_passages.docid: 136409
negative_passages.text: Verify $\int\sec x\ dx=\frac12 \ln \left\lvert\frac{1+\sin x}{1-\sin x}\right\rvert + C$
negative_passages.docid: 54175
negative_passages.text: Prove the trigonometric identity $\sin^4{x} = \frac{3-4\cos{2x}+\cos{4x}}{8}$
negative_passages.docid: 108223
negative_passages.text: If $\cos x=\cos y$ and $\sin x=\sin y$ is $x=y$ or $x=2\pi,y=0$ or $x=0,y=2\pi$? $x,y\in[0,2\pi]$
negative_passages.docid: 75721
negative_passages.text: No. of real solutions of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $
negative_passages.docid: 92898
negative_passages.text: How to integrate $\sqrt{1-\sin 2x}$?
negative_passages.docid: 100777
negative_passages.text: Finding the derivatives of sin(x) and cos(x)
negative_passages.docid: 27132
negative_passages.text: Can't figure out how $\sin\left(\tan^{-1}(x)\right)=\frac{x}{\sqrt {x^2+1}}$
negative_passages.docid: 137060
negative_passages.text: Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
negative_passages.docid: 79684
negative_passages.text: Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$
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: 73445
negative_passages.text: Is Pythagoras the only relation to hold between $\cos$ and $\sin$?
negative_passages.docid: 66872
negative_passages.text: Exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$
negative_passages.docid: 105622
negative_passages.text: How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$?
negative_passages.docid: 119669
negative_passages.text: Evaluate $\int \frac{1}{\sin x\cos x} dx $
negative_passages.docid: 19561
negative_passages.text: Why is $\frac{1}{\frac{1}{X}}=X$?