Quick Inequality Question

Got stuck trying to figure out this proof.

Got here:
sin(x)cos(x) ÷ x < cos(x) < sin(x) ÷ x

Trying to get here:
cos(x) < sin(x)÷x < 1

If I divided both sides by (sin(x)÷x) then won't it turn cos(x) into x*cot(x)? I don't know what to do.

The proof I'm trying to figure out is: lim[x to 0] sin(x)÷x = 1.

sin(x)cos(x)÷x doesn't equal 1, and I don't think it equals sin(2x)÷2x, right? I saw that somewhere and I don't even know how that works. What do I do?? Did I do something wrong? Is it a trig identity or something that I'm missing?

typically you are working with the unit circle , so the measure of the angle = length of

the arc on the circle...thus sin x ≤ x ≤ tan x { we work with x > 0 }

so sin x / x ≤ 1...but x ≤ tan x = sin x / cos x ---> cos x ≤ sin x / x

thus cos x ≤ sin x / x ≤ 1...now squeeze to get the result