Question

proof that tanx + secx -1/tanx - secx+1 = 1+sinx/cosx

Answer 1

The way you typed it, it is false.
It needs brackets, but the number of variations to
place those brackets is too great for me to start guessing
what you meant.
e.g. did you mean :
tanx + (secx -1)/(tanx - secx)+1 = (1+sinx)/cosx
or
(tanx + secx -1)/(tanx - secx+1) = 1+sinx/cosx
or
....