Proof of range of tan $\theta$

Question

I need someone to review the following proof to find the range of tan $\theta$, whether it is acceptable or not:
Starting with |cos$\theta|\le$ 1
Square both sides: |cos$\theta|^2\le 1^2$
Simplify: $cos^2\theta\le 1$
Use reciprocal property for inequality: $\frac{1}{cos^2\theta}\ge\frac{1}{1}$
Use reciprocal identity: $sec^2\theta\ge1$
Use Pythagorean identity: $1+tan^2\theta\ge1$
Simplify: $tan^2\theta\ge 0$
Since the square of any real number is always greater or equal to 0, it follows that the range of tan $\theta$ is the set of all real numbers. ***************************************************************************
I need someone to review the following proof whether it is acceptable or not.
Using the rectangular coordinates definition of $\tan \theta=\frac{y}{x}$ where $\theta$ is in standard position and (x,y) is a point on the terminal side,
tan $\theta=\frac{y}{x}$
=$\frac{y-0}{x-0}$
= Slope of the terminal side (The terminal side starts from the origin)
Since the slope of a straight line is the set of all real numbers, it follows that the range of tan θ is the set of all real numbers.

Answer 1

This is not correct. You deduce, after several steps that $\tan^2\theta\geqslant0$. But this holds for every function. Are you going to say that the range of every function from $\mathbb R$ into $\mathbb R$ is $\mathbb R$?