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?

Here is another method.  The only real disadvantage is that it requires calculating $e^{-1}$ in decimal form, but only two digits are needed, so this can be done by hand easily.  It does not require using very large fractions or remembering the series for $\tan(x)$.
As in another answers, we want to find an upper bound for $\sin(1)$ and a lower bound for $\cos(1)$ so that we can show that $\tan(1) < \frac{\pi}{2} = \sin^{-1}(1)$.  From there, given the monotonicity of $\sin^{-1}(x)$, the result follows easily.
$\sin(1)$ can be expanded as
$$ \sin(1) = \frac{1}{2}\left[ \sin(1) + \sinh(1) \right] + \frac{1}{2} \left[ \sin(1) - \sinh(1) \right] $$
The first half represents the positive terms in the Maclaurin series for $\sin$, and the second term represents the negative terms.  On the second half, just keep the first term:
$$ \frac{1}{2} \left[ \sin(1) - \sinh(1) \right] = - \sum_{n=0}^\infty \frac{1}{(4n+3)!} \le -\frac{1}{6} $$
Plugging this into the first equation and solving for $\sin(1)$ above, we get
$$ \begin{align} \sin(1) &\le \sinh(1) - \frac{1}{3} \\ &= \frac{e-e^{-1}}{2} - \frac{1}{3} \\ &\le \frac{2.72 - 0.36}{2} - 0.333 \\ &= 0.847 \end{align} $$
Now consider the Maclaurin series for $\cos(1)$.  The terms are alternating and decrease in magnitude, so for a lower bound, we can terminate after any negative term.  Keep the first four terms:
$$ \begin{align} \cos(1) & \ge 1 - \frac{1}{2} + \frac{1}{24} - \frac{1}{720} \\ & \ge 1 - \frac{1}{2} + \left( \frac{1}{24} - \frac{1}{600} \right) \\ & = 1 - \frac{1}{2} + \frac{1}{25} \\ &= 1 - 0.5 + 0.04 \\ & = 0.54 \end{align} $$
This gives us
$$ \tan(1) \le \frac{0.847}{0.54} < 1.57 < \frac{\pi}{2} = \sin^{-1}(1) $$
Of course, it would have taken me a really long time to come up with this without the use of a calculator for exploration.
