Derivatives of trig functions

The derivatives of $\sin$ and $\cos$ follow immediately from their
characterization as coordinates (unit speed, counterclockwise)
on the circle: the derivative is the velocity, which is
perpendicular and of unit length.

Alternatively, via power series.
Alternatively, via de~Moivre's formula.

The other derivatives can be derived from these, by quotient rule.

There's not that many to know; the main ones are:
$\sin'=\cos$
$\cos'=-\sin$

Then there's
$\tan' = \sec^2$
$\sec' = \sec \tan$

...then the co-s:
$\cot' = -\csc^2$
$\csc' = -\csc \cot$

A useful mneumonic is that adding a ``co-'' complements everything and
adds a negative; this follows from the substitution
$u=\pi/2 - x$, which switches all the complements and has derivative
$-1$.
(This even works for sine and cosine!)

This is the only way that I can remember the co ones.

It would be good though to give direct geometric reasons for the
others, if for no other reason than fun:
$\tan'$? $\sec'$?
(look at the geometric picture -- ideas?)
