I'm making a video right now about the unit circle definitions of the basic trig functions.
I've done sine and cosine, and am now talking about the tangent function.
As most of you know, it can be rewritten as in the title, but off the top of my head, I can't remember when this might be genuinely useful.
Certain derivatives and integrals come to mind. Do you have any other examples?
You are simplifying the equation for a function and you get down to something like $f(x) = \cos x\tan x $.
You want to know where tan(x) is discontinuous without memorizing it (assuming you already know where cos(x) is zero).
My previous answer (which was basically "That is the definition of tangent!") was not well received. I admit that that answer was made out of exasperation with the question. I have written a longer response as to why I feel the way I do, with some mathematical and historical context.
You say you are defining these functions based on the unit circle. If this is the case, then you are probably defining $\cos(\theta)$ to be the $x$-coordinate, and $\sin(\theta)$ to be the $y$-coordinate of the point obtained by taking a piece of string of length $\theta$, fixing one end at $(1,0)$, and winding it it counterclockwise around the circle. In this case, if you define tangent to be $\frac{y}{x}$ this is also defining tangent to be $\frac{\sin(\theta)}{\cos(\theta)}$. The two should be so closely identified that these two expressions "say" exactly the same thing to you. Because of this I am confused by the question: it appears that the definition of tangent is $\frac{\sin(\theta)}{\cos(\theta)}$.
The unit circle approach takes some care to make rigorous. You at least need to be able to define arclength to make it work. For this reason, most introductions to analysis which I have seen either define sine and cosine in terms of their power series representations, or as solutions to their defining differential equations. In either content, the tangent function would again be defined by the ratio of the two. Trying to define tangent "on its own", without first defining sine and cosine, is difficult because it is not an entire function.
In a certain sense, I am suspicious of the attention given to the other trig functions. We no longer pay homage to haversine or exsecant, yet we still talk about tangent. Why? Fundamentally the reason that tangent, secant, exsecant, etc even have names of their own is that they were needed in some computations quite commonly (say, while sailing), and it was easier to compile tables of values for these "special functions" than to calculate them by hand given only tables for sine and cosine. I do not know the historical reason that only 6 of these functions survived into the common usage, but from a mathematical perspective I feel like only sine and cosine really "deserve" names in the current era. It would be as if we had a new operation "inc sum", denoted $\oplus$ with $a \oplus b = a+b+1$. A useful operation, to be sure, but not useful enough to warrant its own distinct mental real estate.
I also think that the habit of questioning where each and every mathematical fact is "useful" is misguided. Asking about the "usefulness" of an identity which is at most one mental step away from the definition is not a healthy attitude with which to approach mathematics. When someone is learning mathematics, they should always be exploring 3 or 4 steps away from what is written. This is just part of building a mental model, gaining "familiarity with the landscape". Playing with identities to understand the basic relationships between various functions should be appreciated for its own sake. Individual mathematical facts are not useful: a mathematical outlook is useful, a mathematical framework is useful. This particular fact is just one part of building up the mathematical framework of trigonometry.
This is not to say all questions of the form "Examples of when this identity is useful" are not good. For example, I think that the equality expressed by "completing the square" on an arbitrary quadratic is "just an identity", but it has a huge story behind it, and how it connects to the larger story of understanding quadratics. So asking "how is this useful", can prompt an introduction to a whole framework of thinking. Similar comments apply to many identities, like the fundamental theorems of calculus, or Cauchy's formula, etc. $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ on the other hand, is on the same level as $(x+1)(x-1) = x^2-1$ (or even lower on the totem pole): it is just an immediate consequence of definitions. One should not have to justify exploring it. It is not a pivotal part of any story, so it should just be one of those things to observe in passing and appreciate for what it is.
Edit: perhaps the following analogy is useful. If you give someone a blanket, they might ask "What is the use of this particular thread?". The answer to this question for any thread is that, all by itself, it is useless. When you weave all of the threads together you get an object which is incredibly useful. This is the way with mathematics as well. Mathematics, when properly understood, is a powerful framework for understanding many diverse problems. $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ is like one of those threads.
One reason the tangent function is useful is that it yields the slope of a line, $\tan \theta$ where $\theta$ is the angle of a line tangent to a curve at a point (and note the connection of the tangent function to a line tangent to a curve):
More pragmatically, you can use the tangent to estimate the height $y$ of a building: If you are standing about $x$ units from the building, and you need to look up $\theta$ to the top, then $y = x \tan \theta$. Again, this could be explained in terms of $\sin \theta$ and $\cos \theta$.
