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Chapter 5: Problem 26

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. $$\sec ^{4} x-\tan ^{4} x$$

Short Answer

Expert verified
The simplified form of the expression \( \sec^{4} x - \tan^{4} x \) is \( \sec^{2} x + \tan^{2} x \)

Step by step solution

01

Re-write the expression

Rewrite the equation as difference of squares. The expression \( \sec^{4} x - \tan^{4} x \) can be rewritten as \( (\sec^{2} x)^{2} - (\tan^{2} x)^{2} \). This expression can be considered as a difference of squares which can be written in the form \( a^{2} - b^{2} = (a-b)(a+b) \). Here \( a = \sec^{2} x \) and \( b = \tan^{2} x \), yielding the expression to be \( (\sec^{2} x - \tan^{2} x) (\sec^{2} x + \tan^{2} x) \)
02

Apply Pythagorean Identity

Use the Pythagorean identity to simplify the expression. The identity \( \sec^{2} x = 1 + \tan^{2} x \) can be rearranged to \( \sec^{2} x - \tan^{2} x = 1 \). Substitute this into the expression found in Step 1, making the expression \( 1 * (\sec^{2} x + \tan^{2} x) \)
03

Simplify the expression

Thus, the expression simplifies to \( \sec^{2} x + \tan^{2} x \)

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