This gets you almost there: trigcos(tan(x)') yields 1/cos(x)^2 which is obviously sec(x)^2.
See the manual for many other functions that rewrite trig expressions in various ways. However I don't think there are any which rewrite expressions directly in terms of sec(x), csc(x), or cot(x). The students is expected to learn to recognize those by their definitions, I expect.
Disclaimer: I don't work for HP, but HP calculators work for me.