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:

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

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