Compare $\arcsin (1)$ and $\tan (1)$ 
Which one is greater: $\arcsin (1)$ or $\tan (1)$? 

How to find without using graph or calculator?
I tried using $\sin(\tan1)\leq1$, but how do I eliminate the possibility of an equality without using the calculator?

One can write:
$$
\frac{1}{\sqrt{1 + \tan^21}} = \cos1 = 1 - 2\sin^2\frac{1}{2} > \frac{17639}{32768},
$$
because
$$
\sin\frac{1}{2} < \frac{1}{2} - \frac{1}{2^3\cdot3!} + \frac{1}{2^5\cdot5!} = \frac{1920 - 80 + 1}{3840} < \frac{1845}{3840} = \frac{123}{256}.
$$
On the other hand, using Archimedes's lower bound, $\pi > 3\tfrac{10}{71}$:
$$
\frac{1}{\sqrt{1 + \left(\sin^{-1}1\right)^2}} = \frac{1}{\sqrt{1 + \left(\frac{\pi}{2}\right)^2}} < \frac{1}{\sqrt{1 + \left(\frac{223}{142}\right)^2}} = \frac{142}{\sqrt{69893}} < \frac{142}{\sqrt{69696}} = \frac{142}{264} = \frac{71}{132}.
$$
So, one can prove that $\tan1 < \sin^{-1}1$ by proving that:
$$
\frac{17639}{32768} > \frac{71}{132},
$$
which simplifies to $33 \times 17639 > 71 \times 8192$, that is, $582087 > 581632$ - which is true.
