btw , fix your post title ...
So when I did this the proof I got was -tanx and i can't really see where i went wrong:
1+tanx/1+cotx =tanx
(1-cotx/1-cotx) (1+tanx/1+cotx)
1+tanx-cotx-cotxtanx/ 1-cot^2x
(1+(sinx/cosx) - (cosx/sinx) - (cosx/sinx)(sinx/cosx)) / 1-(cos^2x/sin^2x)
((sin^2xcosx- sinxcos^2x/sinxcosx)/( (sin^2x/sin^2x) - (cos^2x/sin^2x)))
((sinxcosx(sinx-cosx)/sinxcosx)) / sin^2x-cos^2x)
sinx-cosx/(sin^2x-cos^2x/sin^2x)
sinx -cosx /1 times sin^2x/(sin^2x-cos^2x)
sin^2x/(sinx-cosx)
- sinx/cosx
-tanx
Hopefully this makes sense ....