query: How to show that $\cos(\sin^{-1}(x))$ is $\sqrt{1-x^2}$?
positive_passages.docid: 5747
positive_passages.text: $\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?
negative_passages.docid: 65923
negative_passages.text: How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
negative_passages.docid: 23288
negative_passages.text: Prove that $x^2<\sin x \tan x$ as $x \to 0$
negative_passages.docid: 97952
negative_passages.text: Square root of 1 is (not) -1
negative_passages.docid: 122145
negative_passages.text: Evaluating Limit - $(1-(\cos x)^{\sin x})/(x^3)$
negative_passages.docid: 143072
negative_passages.text: Proving $|\sin x - \sin y| < |x - y|$
negative_passages.docid: 55436
negative_passages.text: Why does $A\sin{k(x+c)}=a\sin{kx}+b\cos{kx}$ imply that $A=\sqrt{a^2+b^2}$ and $\tan{c}=-b/a$?
negative_passages.docid: 57152
negative_passages.text: Solve $\sin x = 1 - x$
negative_passages.docid: 77130
negative_passages.text: How to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|<2\sqrt{|x-y|}$?
negative_passages.docid: 28125
negative_passages.text: Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
negative_passages.docid: 7296
negative_passages.text: Prove $(1-\cos x)/\sin x = \tan x/2$
negative_passages.docid: 106803
negative_passages.text: How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?
negative_passages.docid: 45619
negative_passages.text: Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$
negative_passages.docid: 139436
negative_passages.text: Show function $f(x,y)=(x^2-y^2,2xy)$ is $1$-$1$ by Inverse Function Theorem
negative_passages.docid: 46353
negative_passages.text: Proof that $\cos(\pi/3)=1/2$
negative_passages.docid: 84420
negative_passages.text: When is $\sin(x) = \cos(x)$?
negative_passages.docid: 36063
negative_passages.text: Finding the limit of $(1-\cos x)/x^2$