Why do we need so many trigonometric definitions?
There are a lot of trigonometric functions which are defined geometrically, which we rarely use anymore. Many of these are summarized by this image:
These all have their uses in particular circumstances. For example, the half versed sine (or haversine) is useful for determining the great circle distance between points, which is incredibly useful if you are trying to navigate. We don't need the haversin, but it is useful, and reduces notation a bit in at least one specific context. The other trig functions are similar—personally, I would rather write $$ \frac{\mathrm{d}}{\mathrm{d}t} \tan(t) = \sec(t)^2 $$ than $$ \frac{\mathrm{d}}{\mathrm{d}t} \frac{\sin(t)}{\cos(t)} = \frac{1}{\cos(t)^2}. $$
EDIT: This answer was written when the question seemed to be asking about the "necessity" defining secant and cotangent functions. It seems that the original questioner had a much more general question in mind, i.e. why do we need any definitions at all? The only possible response that that, I think, is because mathematics would be impossible without "definitions." Working under the assumption that the original questioner is in earnest, a partial answer is as follows:
A huge part of mathematics is the language we use in order to communicate mathematical ideas. We could, I suppose, never define anything beyond the basic axioms, but then we could never get anything done, and would have no hope of ever communicating our ideas to others. If we don't define a derivative, how do we describe the the motion of a planet? It would be cripplingly inconvenient if we could never write $3$, and always had to write $\{ \{\}, \{\{\}, \{\{\}\} \}, \{\{\}, \{\{\}, \{\{\}\}\} \}$. Not only is that quite hard to read (do you really want to check that I got all of my commas and braces right?), it is horribly inefficient. And this is just to describe a relatively small natural number. It only gets worse from here!
The point is that definitions allow us to encapsulate complicated ideas into a short collection of symbols (i.e. words) that allow us to make further deductions. Definitions are at the very heart of mathematics. We can do nothing without them.
You could equally ask why we define "$8$" when we could just write $1+1+1+1+1+1+1+1$.
It's convenient to have shorter names for things that get used a lot. Admittedly, "$8$" is much more convenient (and much more often used) than something like $\sec$, so there's always room to argue about whether a particular abbreviation is really useful.
The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)}$ etc. is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say $\sec(x)$ values than calculate $\frac{1}{\cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.
I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown (Tattoos On Math) and the other is from an amazing guy called Simon Clark. (Why $\sin$ and $\cos$ don't mean anything).
Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest can be derived. But the world is sometimes a real scary place without $\tan(x)$, $\cot(x)$, $\sec(x)$ and $\operatorname{cosec}(x)$.