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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 35 (page 474)

In problems 19-98, establish each identity.
$\sec^{4} (胃) - \sec^{2} (胃) = \tan^{4} (胃) + \tan^{2} (胃)$

Short Answer

Expert verified

We have proved the property $\sec^{4} (胃) - \sec^{2} (胃) = \tan^{4} (胃) + \tan^{2} (胃)$ .

Step by step solution

01

Step 1. Given information.

We have:

$\sec^{4} (胃) - \sec^{2} (胃) = \tan^{4} (胃) + \tan^{2} (胃)$

02

Step 2. Establish the identity.

Take left hand sides:

$\sec^{4} (胃) - \sec^{2} (胃)$

Break:

$= \sec^{2} (胃) \sec^{2} (胃) - \sec^{2} (胃)$

Taking common out:

$= \sec^{2} (胃) (\sec^{2} (胃) - 1)$

Apply identity: $s e c^{2} 胃 = 1 + \tan^{2} 胃$ :

$= (1 + \tan^{2} (胃)) (\tan^{2} 胃) = \tan^{2} 胃 + \tan^{4} 胃$

Hence, proved.

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