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 $sin^2x$ .

How do you simplify (sec(x))^2−1(sec(x))^2−1?