What is the deepest / most interesting known connection between Trigonometry and Statistics?

No doubt it's the Law of Cosines. The correlation between two data sets follows the generalized $n$-dimensional Law of Cosines.

EDIT: Maybe I'll make this a little more explicit. Take two data sets $A = (5,7,2...)$ and $B = (12, 4, 9...)$ and ask if they are correlated. One way is to treat them as vectors and look at the data set $C = A+B = (17, 11, 11...)$ where the sum of the data sets (vectors) is taken pointwise. Okay...it's not the length of the vectors that works like the Law of Cosines, but the standard deviation. If the two data sets are randomly correlated then you should expect the standard deviations to add like the Law of Pythagoras, so that if $\text{StDev}(A) = 3$ and $\text{StDev}(B)=4$ then $\text{StDev}(C)$ should equal $5$. For $100\%$ correlations, the standard deviation of $C$ would have to be $7$ (or $1$ for negative correlation). It's the Law of Cosines where the correlation is the cosine of the angle between two vectors of length $3$ and $4$.

The normal distribution has a $\pi$ in it. That's fairly deep.

Edit: I don't see how this doesn't count. $\pi$, after all, is $4 \arctan 1$, or half the period of the $\sin$ and $\cos$ functions. It is closely related to trigonometry and the properties of the trigonometric functions, and to call $\pi$ an occurrence of "geometry" as if that were something unrelated to trigonometry is mystifying.

Perhaps this would be quite difficult to include in a high school class, but wouldn't Fourier analysis be a good example of this?

The characteristic function of a random variable which admits a density is just the Fourier transform of its density, and Fourier transforms are continuous versions of Fourier series which involve decomposition into sines and cosines. More explicitly, Fourier transforms involve exponentials of purely imaginary numbers which could also be written as trigonometric functions.