Fundamental Theorem of Trigonometry

$$\boxed{\sin^2x+\cos^2x=1.}$$

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.

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) ).