How to simplify (sec⁡(x))2−1?

Question

Royce Mathis · Accepted Answer

This is a use of the Pythagorean identities, specifically:

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

\frac{\cos^{2} x}{\cos^{2} x} + \frac{\sin^{2} 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

.

How to simplify (sec(x))^2−1?