Fundamental Theorem of Trigonometry This is a pretty open ended question and I apologize, in advance, if this is not the place for it.  But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and why?  Should we have to restrict ourselves to the planar case... I think so.
 A: $$\boxed{\sin^2x+\cos^2x=1.}$$
A: Fundamental theorem, imho, would be:
A magnifying glass that increases the size of an object $k$ times:
1) Doesn't change angles
2) Increases length by a factor of $k$
From this you can (informally) derive the existence of sine, cosine, $\pi$, the $k^2$ increase in area, figure out problems of similar triangles, etc.
A: Answer: The Fundamental Theorem of Trigonometry is

In a unit circle, an arc of length $2x$ stands on a chord of length $2sin(x)$.

Source: Goodstein's Mathematical Analysis
Argument: This theorem connects the geometric definition of the trig functions with the analytic definition of the trig functions.
Proof: The points $(sin(\alpha), cos(\alpha))$ and $(-sin(\alpha), cos(\alpha))$ with $0 \leq \alpha \leq \frac{1}{2}\pi$ are the endpoints of a chord on the unit circle $x^2+y^2=1$ having length $2sin(\alpha)$. The length of the arc joining them is
$$
\int_{-sin(\alpha)}^{\sin(\alpha)} \sqrt{1 + \left(\frac{dy}{dx}\right)^2}dx = \left[arcsin(x)\right]^{sin(\alpha)}_{-sin(\alpha)} =2 \alpha
$$
Remark: The argument that the integral is equal to $2\alpha$ uses the definition of sine as the limit of a power series (or whatever analytic definition you feel is most appropriate), but the coordinates come from the definition of sine as the x-coordinate of the point of intersection of a line through the center of a unit circle.
Note: This proof can be found on page 166-167 of Goodstein's Mathematical Analysis.
(Additionally one can see this theorem as fundamental in that it connects the ancient development of trigonometry to our modern use of the subject: https://en.m.wikipedia.org/wiki/Chord_(geometry) ). 
A: The identity $\sin^2x+\cos^2x=1$ comes from Pythagoras. 
I think the fact that $\sin x$ and $\cos x$ (for a right-angled triangle) are well defined at all is the fundamental theorem.

Fundamental Theorem of Trigonometry The ratio between corresponding sides of similar triangles are equal.

Edit: Daniel V has a similar idea.
A: It was recognized by the 1800s that the angle difference formula for cosine 
$$\cos(a-b) = \cos a \cos b + \sin a \sin b$$
is fundamental in the sense that the other relations of trigonometry can be derived from it using a functional equation approach. See Aczel on this.
Also, though people often believe the contrary, $\cos^2 x + \sin^2 x = 1$ for acute angles can be derived from the angle difference formulas independently of Pythagoras. (See my article in Forum Geometricorum.)
A: $e^{i\theta}=\cos \theta + i \sin \theta$.
You can derive a bunch of the relations from here!
A: My attempt: a right triangle with unit hypothenuse and angle $x$ has base $\cos x$.
A: $$\frac{\sin x}{\cos x} = \tan x$$
$$sin^2 x + cos^2x = 1$$
