# How do you simplify (sec(x))^2−1?

Sep 7, 2015

Using the Pythagorean identity:

${\tan}^{2} x = {\sec}^{2} x - 1$

#### Explanation:

This is an application of the Pythagorean identities, namely:

$1 + {\tan}^{2} x = {\sec}^{2} x$

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

${\cos}^{2} x + {\sin}^{2} x = 1$

${\cos}^{2} \frac{x}{\cos} ^ 2 x + {\sin}^{2} \frac{x}{\cos} ^ 2 x = \frac{1}{\cos} ^ 2 x$

$1 + {\tan}^{2} x = {\sec}^{2} x$

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

${\tan}^{2} x = {\sec}^{2} x - 1$

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

${\cos}^{2} x + {\sin}^{2} x = 1$

$1 + {\tan}^{2} x = {\sec}^{2} x$ (divide all terms by ${\cos}^{2} x$)

${\cot}^{2} x + 1 = {\csc}^{2} x$ (divide all terms by ${\sin}^{2} x$)

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