I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.
Is there any good examples of their uses outside academia?
[Math] Real world uses of hyperbolic trigonometric functions
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Homotopy theory / algebraic topology was born out of applications rather than abstract nonsense considerations. So there's plenty of applications, as that's how the subject began.
Perhaps the first topological proof would be the bridges of Konigsberg problem: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
Where algebraic topology started getting off the ground was in the work of Poincare. The Poincare-Hopf Index theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem
was a landmark. In its natural setting it was a relationship between Euler characteristic, tangent bundles and intersection theory. But from the perspective of a differential equator it's a fundamental tool that allows you to determine whether or not differential equations have fixed points.
Applications have piled-up over the years. Some of the more modern ones are listed in other people's responses. The birth of topological dynamics in the mid 20-th century was of course a big one.
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.
Best Answer
If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve.