How do you simplify #(sec(x))^2−1#?

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Answer 1 · 2015-09-07T08:00:14

Using the Pythagorean identity:

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

This is an application of the Pythagorean identities, namely:

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

This can be derived from the standard Pythagorean identity by dividing everything by #cos^2x#, like so:

#cos^2x + sin^2x = 1#

#cos^2x/cos^2x + sin^2x/cos^2x = 1/cos^2x#

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

From this identity, we can rearrange the terms to arrive at the answer to your question.

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

It would help you in the future to know all three versions of the Pythagorean identities:

#cos^2x + sin^2x = 1#

#1 + tan^2x = sec^2x# (divide all terms by #cos^2x#)

#cot^2x + 1 = csc^2x# (divide all terms by #sin^2x#)

If you forget these, just remember how to derive them: by dividing by either #cos^2x# or #sin^2x#.