TITLE: Why do we need so many trigonometric definitions?
QUESTION [54 upvotes]: Examples
$$\sec(x) = \dfrac{1}{\cos(x)}$$
$$\cot(x) = \dfrac{1}{\tan(x)}$$
There are many more out there, but why do we need definitions that can be written with just $\sin , \cos ,\tan $ etc. in maths? Why can't they just be written as their expanded form?
Most trigonometric functions can be written with just $\sin \cos$ and $\tan$. Why do we need so many?
Additionally, I mean all definitions.

REPLY [5 votes]: The tangent comes back (wearing a different hat) in calculus.  However, you need these definitions.  Let's pick one example.
Suppose we only know $\cot \theta = \frac{1}{\tan \theta}$.  Then cotangent is undefined whenever tangent is undefined or zero, which is every integer multiple of $\pi/2$.  But if we know $\cot \theta = \frac{\cos \theta}{\sin \theta}$, we find out cotangent is only undefined when sine is zero, which is every even integer multiple of $\pi/2$.  
This is practice for a pattern that happens over and over -- different definitions of a function agree where they are both defined, but may each be defined in places where the others are not.  In fact, there is a reasonable chance you have already done this twice with sine and cosine (and not really noticed because it might not have been pointed out).  Originally, you only had trig functions defined for acute angles in geometric triangles -- only defined for angles $\theta$ with $0 < \theta < \pi/2$.  Then you extended these to be defined on the unit circle for $0 \leq \theta \leq 2\pi$ by using the $\sin \theta = y, \cos \theta = x, \tan \theta = y/x, \dots$ definitions.  Then you extend these to all real angles $-\infty < \theta < \infty$ by observing coterminal angles all intersect the unit circle at the same point.  That's three definitions -- one using geometric triangles, one using coordinates on a unit circle, and one using coterminal angles to extend from a definition with a small domain to a larger domain.
This is a common activity.  Limits of indeterminate forms are attempts to extend the domain of the difference quotient to the derivative and to extend the domain of (finite) Riemann sums to the integral.  Analytic continuation is more of this.  It goes on and on.
The fundamental lesson is that there is some sort of Platonic ideal abstract function, but each recipe we can find to evaluate it only tells us about the values on some subset of its domain.  Different definitions cover different domains.  To really be able to work with a function you need to be able to go every it can go -- not just everywhere one recipe for it can go.