Is it possible to deduce that $\sin(x)< x$ for $0x$? It is not difficult to individually prove that in the domain $0<x<\pi$, we have 
$$\tan x > x \quad(A)$$
and 
$$\sin x < x \quad (B)$$
However, I was wondering if it was possible to get result $(B)$ for free from $(A)$, or possibly visa versa.

I would do the following:
if $x>0$ then $x=\int\limits_{0}^x1dt\geq \int\limits_{0}^{x}\cos(t)dt=\sin(x)$
