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I think we need to take limit.. but... we have something to do before taking limit?

Well, I don't think we need integration here.

Ohh Ok. There are some proof of function being greater than the other.. hmm

Well, that's the problem.. How are you going to show it? Finding its derivative?

you cannot just say that the roots are at n*pi?

How so?

You want a proof of that as well...

I mean x= Tan x is a pain.....

Yes!

You just have do a graph y=x and Tan x and look or use a CAS

Graphing... is not that good...

If graphing works, why not just graph the function sinx / x to show it's a decreasing function?

Well, quite...:-)

What are you doing, Fourier?

No.. limit and derivative.

That's a bit sneaky, hitting you with sinc function...

Let's see , all you have to do is show that the first positive root is at pi....

Sin pi/pi = 0

And anywhere from 0 to pi is not 0

That's good enough to show the first root is at pi, right?

Oh... so you're solving sinx/x=0? If so, yes!

So now you are done, I think (assuming usual smoothness principles).

Wait.. how.. why...
xcosx - sinx = 0
xcotx -1 = 0
xcotx = 1
Am I doing anything wrong here?

Don't think while you eat!

I know I have seen a proof that tanx>x in (0,pi/2) ... that would work if I could remember it

This is complicated...

Yes!

okay, we can prove this using the mean value theorem...

the mean value theorem states that f(x)=f(a)+f'(c)(x-a) for some a

getting what we need from the formula
f(0)=0, f'(c)=sec^2(c) where 0

what do you get then by using this in the formula?

f(x)=f(a)+f'(c)(x-a)
tanx = 0 + sec^2 (c) (x)
tanx = [sec^2 (c)] x
Doesn't look good.

it is though, because what do we know about sec(c) for 0

0

yeah, but the point is that sec(c)>1 for that interval

so
tanx=sec^2(c)x 0

1< sec^2(c)
x < sec^2 (c) x
x< tanx
Wow!!

tadah! :)
so that completes your proof

Uh-huh! Why did you think of mean value theorem for this question????

It's the only way I know to prove tanx>x in (0,pi/2)
there may be other ways

very similar argument^

no apologies necessary, always good to see more than one explanation :)

Night!!

See ya !