Answer
See below.
Work Step by Step
1. To show that the range of the tangent function is the set of all real numbers, we need to show that for any given value $a\in R$, we can find an angle such that $tan\theta=a$.
2. Use the unit circle definition, we have $tan\theta=\frac{y}{x}$ where $x^2+y^2=1$.
3. Use the relation $y=\pm\sqrt {1-x^2}$, we have $tan\theta=\frac{\pm\sqrt {1-x^2}}{x}$.
4. For any given value $a\in R$, let $tan\theta=\frac{\pm\sqrt {1-x^2}}{x}=a$, we have $a^2x^2=1-x^2$, thus $x=\pm\sqrt {\frac{1}{a^2+1}}$.
5. The above shows that for any given value $a\in R$, we can find a pair of $(x,y)$ values given above such that $tan\theta=a$ implying that the range of $tan\theta$ is all real numbers.
