# Proving that any real number can be expressed as a ratio of $2$ real numbers in $[-1,1]$

I got this question while thinking about trigonometry.

We know that $$\cos x \in[-1,1]$$ and $$\sin x \in[-1,1]$$.

We also know that $$\tan x \in \Bbb R$$.

For any real number $$y$$, we can have a value of $$x$$ such that $$y=\tan x=\frac{\sin x}{\cos x}$$. This is easy to see, as we can theoretically always draw a triangle having leg lengths as $$y$$ and $$1$$.

But can we prove this without using trigonometry?



So the formal statement of the question would be (I made it up; it may/may not be a real question):

Prove that all real numbers can be expressed in the form of $$\frac{p}{q}$$, where $$p,q\in[-1,1]$$ and $$q\neq0.$$

Your trigonometric insight reduces to a fact not mentioning its inspiration,$$t=\frac{\frac{t}{\sqrt{1+t^2}}}{\frac{1}{\sqrt{1+t^2}}}.$$
Hint: If $$x \notin [-1,1]$$, then $$1/x \in [-1,1]$$.