I'm doing this completely from memory, and for no purpose whatsoever.

Well, okay, I do have to memorize 20 trig identities before my test tomorrow (easy points on a hard test), but I could just as easily do this on paper.  I don't know why I'm not.
sin(x)=1/(csc(x))
csc(x)=1/(sin(x))
cos(x)=1/(sec(x))
sec(x)=1/(cos(x))
tan(x)=(sin(x))/(cos(x))
tan(x)=1/(cot(x))
cot(x)=(cos(x))/(sin(x))
cot(x)=1/(tan(x))
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
sin(a-b)=sin(a)cos(b)-cos(a)sin(b)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
cos^2(x)+sin^2(x)=1
tan^2(x)+1=sec^2(x)
cot^2(x)+1=csc^2(x)
cos^2(x)=(1+cos(2x))/2
sin^2(x)=(1-cos(2x))/2

Grr.  I only remember 17.  "Back to the old drawing board."
^Not an appropriate quote, but deal with it.