If you have the trigonometric identity tanx+cotx = secxcosecx
this can be verified by rewriting tanx and cotx in their sinx and cosx forms, getting a common denominator etc.
However, if you were to go down the root of tanx + 1/tanx
tan^2x/tanx + 1/tanx
(tan^2x+1)/tanx
sec^2x/tanx
however, can this be simplified to give secxcosecx or have I made an error I haven't seen?