How may one show that $|\sin x-\sin y|\le |x-y|$?
49054
Show that $\left | \sin a - \sin b \right | \leq \left | a-b \right |$
127759
Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
39082
Proving $x^\alpha-\alpha x \le 1- \alpha $
29679
Prove: $|x-y|\leq |x|+|y|$
108704
Help with proof of $|x+y| \geq |x|-|y|$
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
24311
Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
150819
How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
126701
How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule?
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
126165
Prove $|\arctan x - \arctan y|\leq |x-y|$
133880
$|\sin^2x-\sin^2y|\le K|x-y|$ find K
95266
Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$
44119
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
18967
Linear independence of $\sin(x)$ and $\cos(x)$
42976
How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$
73047
Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$