how do you prove

cosx+sinxtanx=secx

The usual way to prove any of these trig identities is to convert everything to sines and cosines and simplify. Sometimes you have to be cleaver.

cos(x) + sin(x) tan(x) = sec(x)

identity: tan(x) = sin(x) / cos(x)

cos(x) + sin(x) * sin(x)/cos(x) = 1/cos(x)
cos(x) + sin^2(x)/cos(x) = 1/cos(x)
[cos^2(x) + sin^2(x)] / cos(x) = 1 / cos(x)

identity: sin^2(x) + cos^2(x) = 1

identity: sec(x) = 1/cos(x)

sec(x) = sec(x)

quod erat demonstrandum