Years ago I found this identity through experimentation playing with data and transformations. After explaining it to my statistics professor he came in the next class with a one-page proof using vector and matrix notation. Unfortunately I lost the paper he gave me. (This was back in 2007)
Is anyone able to reconstruct a proof?
Let $(x_i,y_i)$ be your original data points. Define a new set of data points by rotating the original set by angle $\theta$; call these points $(x'_i,y'_i)$.
The R squared value of the original set of points is equal to the negative product of the derivative with respect to $\theta$ of the natural log of the standard deviation for each coordinate of the new set of points, each evaluated at $\theta=0$
$r^2= - \left(\left.\frac{d}{d\theta}\ln(\sigma_{x'})\right|_{\theta=0} \right) \left(\left.\frac{d}{d\theta}\ln(\sigma_{y'})\right|_{\theta=0} \right)$