TITLE: Proving that $\sin x \ge \frac{x}{x+1}$
QUESTION [16 upvotes]: Prove that
$$ \sin x \ge \frac{x}{x+1}, \space \space\forall x \in \left[0, \frac{\pi}{2}\right]$$

REPLY [2 votes]: Inequality is easily verified for $x=0,1$
$\csc x - \cot x = \frac{1}{\csc x + \cot x}<1 $ for $x \in \left(0, \frac{\pi}{2} \right)$
Hence, $\csc x < 1 +\cot x < 1+ \frac{1}{x} $ since its well known that $\tan x>x$
and this leads to the desired inequality