I understand how to prove that using the definitions of the elementary trig functions:
$$\frac{\sin\theta}{\cos\theta} = \frac{\dfrac{opp}{hyp}}{\dfrac{adj}{hyp}} = \frac{opp}{adj} = \tan\theta$$
However, I have a hard time visualizing that identity. Is there a way to do so?
There is a very nice way to visualize these functions. First, you need to draw the unit circle and pick some point on it. $\theta$ will be the angle between the point (1,0) anti-clockwise towards the point you chose. Then $\cos\theta$ is the $x$-coordinate and $\sin\theta$ is the $y$-coordinate of the point. Now, draw the vertical line to the $x$-axis going trough the point $(1,0)$. Moreover, draw the line going through your point and the origin. Mark the intersection of this line and the vertical line you drew before. The $y$-coordinate of this point is $\tan\theta$. That also tells you what happens when $\theta$ approaches $90$ degrees or $270$ degrees! Try to understand why this is true! If you have more questions about it, write a comment.