TITLE: Is there an integral that proves that $\sin \tan 1\lt 1$?
QUESTION [5 upvotes]: I recently noted that this inequality is unbelievably sharp:
$$\sin \tan 1\lt 1$$
Is there some sort of integral that can prove that this is true?
This question might be of some use:
Prove: $\sin (\tan x) \geq {x}$
I also noted, after seeing some of the comments, that an integral of $$\pi/2-\tan 1$$
would also greatly help

REPLY [5 votes]: Hint: 

$\sin(x) \lt 1$ unless $x = \frac{\pi}{2} + 2n\pi$ for integer $n$