Simplifying an expression again

https://www.quora.com/profile/Bravewarrior/p-152921930
I am stuck on the last two lines of working. I managed to get to the third last line, but am struggling to simplify my answer to what they got in the mark scheme in the end. Any help would be great!

The usual
(a+b)/c = a/c + b/c
and 1/cos = sec.

For some reason I am still confused. How did they get to e^tanx (sec^3 x +secxtanx)? 😭😭

sec/cos^2 =sec*(1/cos)^2 = sec^3
sin/cos^2 = (sin/cos)*(1/cos) = tan*sec

Note if you did it like the other post so
e^tan / cos = e^tan (cos)^(-1)
then product rule gives
sec^2 e^tan / cos + sin e^tan / cos^2 = sec^3 e^tan + sec * tan e^tan = ...
Slighlty simpler but not a huge amout