Is there an integral that proves that $\sin \tan 1\lt 1$?

Question

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

Answer 1

Hint:

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