How do you simplify #(tanx + secx)(tanx - secx)#?

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Answer 1 · 2016-06-27T01:03:25

Do FOIL and simplify to get to #-1#

Let's start with the original:

#(tanx+secx)(tanx-secx)#

Let's now do FOIL:

#(tanx)(tanx)=tan^2x#
#(tanx)(-secx)=-tanxsecx#
#(secx)(tanx)=tanxsecx#
#(secx)(-secx)=-sec^2x#

Adding them all up:

#tan^2x-tanxsecx+tanxsecx-sec^2x=tan^2x-sec^2x#

There is a trig identity:

#tan^2x+1=sec^2x#

#tan^2x-sec^2x=-1#

So we can finish up our simplification:

#tan^2x-tanxsecx+tanxsecx-sec^2x=tan^2x-sec^2x=-1#