Here are two answers, each based on a different definition.
Answer 1: We define, for acute angles with one ray on the positive $x$-axis and the other in the first quadrant, the sine and cosine as the $y$ and $x$ coordinates of the point where the second ray meets the unit circle. This gives us a notion of sine and cosine of $x$ for $0 \le x \le \pi/2$.
For obtuse angles $\pi/2 < x \le \pi$, we define $\sin(x) = \sin(\pi - x)$ and $\cos(x) = -\cos(\pi - x)$. We verify (this takes a little work) that sine and cosine, even for these new angles, satisfy $\sin^2(x) + \cos^2(x) = 1$, and that they're continuous at $x = \pi/2$, and that the two definitions agree there.
[Let me do that work: suppose that $\pi/2 < x /le \pi$; then $\sin(x) = \sin(\pi-x)$, where the latter argument is a number $y = \pi - x$ between $0$ and $\pi/2$; similarly, $\cos(x) = -cos(y)$. So $\sin^2(x) + \cos^2(x) = [\sin(y)]^2 + [-\cos(y)]^2 = \sin^2(y) + \cos^2(y)$, where $y$ is an acute angle, so this sum is $1$.]
We then do something similar for angles between $\pi$ and $3\pi/2$ (defining $\sin(x) = -\sin(x-\pi), \cos(x) = -\cos(x-\pi)$), and for angles between $3 \pi/2$ and $2\pi$ (I'll let you fill in the definition yourself).
Then we define sine and cosine for an arbitrary angle $a$ in any triangle $ABC$ by observing that there's a unique translation and rotation that moves the ray $AB$ to the positive $x$-axis, and the ray $AC$ to ...some ray starting at the origin. You can then use the earlier definition of sine and cosine to define the sine and cosine of $a$. And that finishes things up.
Of course, proving the existence an uniqueness of a translation $T$ and rotation $R$ with the property that $R(T(A)) $ is the origin, etc., involves some real work, which is why I hate this approach.
Approach 2: There are, on the rationals, two unique functions, $s$ and $c$, with the following four properties:
$s(0) = 0; c(0) = 1; s(\pi/2) = 1.$
For $0 < x < \pi/2$, $0 < s(x) < x$.
$c(a-b) = c(a) c(b) + s(a) s(b)$ for all rational $a, b$.
$s$ and $c$ are continuous on the rationals.
[I think I have those right, but I might be mistaken; I'm pretty sure they're in Apostol's Calculus book]
Proving the existence and uniqueness of these two functions requires some real work. You basically use formula 3 to derive a cosine double-angle formula, and then show that the facts in item 1 (plus formula 3) let you compute sines and cosines of things like $\pi/2, \pi/4, 3\pi/4, \pi/8, ...$, basically, $\pi$ times any rational whose denominator is a power of two. Then you use a limit argument (this relies heavily on item 2) to conclude that the value on all rationals must be determined because of property 4.
THEN, you say "every continuous function on a dense subset of the reals extends to a unique continuous function on the reals" (which requires proof, and a notion of "dense"!), and you say that we'll call the unique continuous extension of $s$ by the name "sine", and the extension for $c$ by the name "cosine". Finally, plugging in $a = b = x$ for any rational $x$ in formula $3$, you see that
\cos(0) = \cos(x)^2 + \sin(x)^2,
and since, by rule 1, we know that $\cos(0) = 1$, we get that the fundamental identity holds on the rationals, which is to say, the function
u(x) = \cos^2(x) + \sin^2(x) - 1
is zero for every rational $x$. It's also continuous, so it has a unique continuous extension to the reals, namely, the everwhere zero function. So for every REAL number $x$, we have
0 = \cos^2(x) + \sin^2(x) - 1
i.e., the fundamental identity is proved.
Of course, although the mathematics here is not terribly difficult, it's really not immediately clear where the four "axioms" came from, or why the functions $s$ and $c$ have anything to do with the things we're used to calling "sine" and "cosine".
There's one more version of this that's possible, which involves defining sines and cosines using matrices (or complex numbers), but I'm not sure that approach is likely to be much more enlightening than these.
The short summary is "This is actually somewhat challenging to do rigorously, which is why basic courses tend to gloss over it."