Consider the following figure:

Compare the area of the triangle with the area of the sector to prove

Further, prove,

* Proof. * Assume the radius is . For we have the triangle has base length and height the area is

The area of the circular sector is

Since the area of the triangle is less than the area of the sector we have,

Then, since we have for

for

, but since the in the specified range we get , as a result, it doesn't justify . Of course, with its very easy to solve.

Hi, thanks! Yes, that was definitely incorrect. I put up a fix now, and also corrected the typo in the diagram that had the origin labelled with $P$.

but we assume 0<x<pi/2 this would imply sinx<x for 0<x<pi which is not true

Your implication is false. x < pi/2 doesn't imply that pi/2 < x < pi, as you are implicitly stating by holding sin(x) < x for higher values of x.

It's like if I say "0 + x < 10 for 0 < x < 9" and you answer with "but this would imply that 0 + x < 10 for 0 < x < 548392 which is not true". Review your logic.