Expert: Paul Klarreich - 1/20/2011

Question
I am studying trigonometry in my pre-calculus class and we're learning about simplifying trigonometric expressions using identities. The problem I'm having trouble on is this one:
(1+sinx)(secx-tanx)
To start, I foiled the expression to get: secx-tanx+sinxsecx-sinxtanx, but I am not sure how to simplify the problem further. I think you have to do something with the "sinxsecx" in order to get them canceled out, but when I tried using the reciprocal identities I didn't see how anything would cancel.

Thanks for any help!

Meghan

Answer
Questioner: Meghan
Country: United States
Category: Calculus
Private: No
Subject: Trigonometric identities question
Question: I am studying trigonometry in my pre-calculus class and we're learning about simplifying trigonometric expressions using identities. The problem I'm having trouble on is this one:
(1+sinx)(secx-tanx)
To start, I foiled the expression to get: secx-tanx+sinxsecx-sinxtanx, but I am not sure how to simplify the problem further. I think you have to do something with the "sinxsecx" in order to get them canceled out, but when I tried using the reciprocal identities I didn't see how anything would cancel.

Thanks for any help!

Meghan
....................................

================ NOTATION ==========================
I write   s = sin x,  c = cos x, to save typing.
====================================================

(1 + sinx)(secx - tanx) =


(1 + s)(1/c - s/c) =  << recip and quotient
       1 - s  
(1 + s)(------) =  << common denom
         c

1 - s^2  
(--------) =  <<< now multiply out. (we don't say 'foil' here)
   c

c^2              <<< pythag identity
--- = cos x
c

Cute.  (I'm sure you are, too.)