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

Question

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?

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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 $q\neq0.$

Answer 1

Your trigonometric insight reduces to a fact not mentioning its inspiration,$$t=\frac{\frac{t}{\sqrt{1+t^2}}}{\frac{1}{\sqrt{1+t^2}}}.$$

Answer 2

Hint: If $x \notin [-1,1]$, then $1/x \in [-1,1]$.