Expert: Socrates - 11/20/2006

Question
This is a problem that has had me stuck for about 4 days now. It's a take home test and it is due tuesday! I've (of course) tried a number of ways to work it, but this is the best thing I've found. I'll also show you which phrase has me stuck. The identity to be proven:

(cosx/1+sinx) + tanx = sec x

i tried to convert everything first to sin or cos. I ended up with:
  [(cos^2x / cosx) + sinx(cosx)] + [(sin^2x + sinx)/ cosx +sinx(cosX)]= sec x
and that's as far as I could go. I think it's the 1+cosx that is really giving me trouble. It doesn't feel right. If you could lead me in the right direction I would greatly appreciate it!

Answer
Just multiply the numerator and denominator of
cosx/(1+sinx) by 1-sinx , this gives

(cosx)(1-sinx)/(1-sin^2x) =

(cosx)(1-sinx)/cos^2x  =

(1-sinx)/cosx  =

1/cosx - sinx/cosx =

secx - tanx  

so , you now have

(cosx/1+sinx) + tanx = secx - tanx  + tanx = secx