Geometric intuition for derivatives of basic trig functions I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, \tan, \sec, \csc, \cot$):

I was initially a bit skeptical about how easy it might be, but then I found this very simple proof for $\sin$ and $\cos$; the basic insight can be seen in this picture from an alternative version of the proof I found later:

Basically, we use the fact arc $PQ$ and segment $PQ$ are the same as $\Delta\theta\rightarrow 0$, and the former has measure $\Delta\theta$.
Nevertheless, I've had no luck so far getting a proof for $\sec$; I have a feeling the proofs of $sec$ and $tan$ are very closely related, as are the $\csc, \cot$ proofs. 
Has anyone seen a proof for the four remaining basic functions anywhere? Perhaps I just haven't drawn the right picture yet.
Another possible avenue is this representation:


This is a direct corollary of @JimBelk's wonderful answer, but I thought it might be nice to have a concise summary graphic which gets all the derivatives at once, which it turns out is pretty simple to do from the second unit circle picture I posted with my question:

The blue triangle shows the derivatives of $\sin\theta$ and $\cos\theta$, the yellow triangle shows the derivatives of $\sec\theta$ and $\tan\theta$, and the salmon triangle shows the derivatives of $\csc\theta$ and $\cot\theta$.
It should be straightforward to verify this using similar triangles and a limit argument to show the green segments are equal in length to the arcs of 3 circles of appropriate radii.
