how would you determine a formula for cot(2x) using only cot(x)? Thenhow would you determine a formula for sec(2x) using only sec(x)?

Answer 1

cot 2x = 1/tan 2x
= 1/[2tanx/(1-tan^2(x))]
= (1-tan^2(x))/2tanx
= [1 - 1/cot^2(x))/(2/cotx)
= [(cot^2(x) - 1)/cot^2(x)]/[2/cotx]
= (cot^2(x) - 1)/(2cotx)

I don't know if that is the simplest way, I just sort of followed by nose.

you can try sec 2x by noting
sec 2x = 1/cos 2c
then using cos2x = 2cos^2(x) - 1

follow the above procedure.