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Chapter 5: Problem 68

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 expression \( \sec ^{4} x - \tan ^{4} x\) simplifies to \( \sec^{2}x + \sec^{2}x*\tan^{2}x\)

Step by step solution

01

Factorize using difference of squares

Identify that the expression can be factored using the difference of squares: \(a^{4} - b^{4} = (a^{2} + b^{2})(a^{2} - b^{2})\). Thus, \( \sec ^{4} x - \tan ^{4} x = (\sec^{2}x + \tan^{2}x)(\sec^{2}x - \tan^{2}x)\)
02

Simplify using identities

Replace \(\sec^{2}x - \tan^{2}x\) with 1, as this is one of the fundamental trigonometric identities. In the same expression \(\sec^{2}x + \tan^{2}x\) can be rewritten as \(\sec^{2}x (1+\tan^{2}x)\). Now, our expression will be simplified to \(1*(\sec^{2}x (1+\tan^{2}x))\)
03

Distribute the terms

Perform the distribution, and the expression simplifies to \( \sec^{2}x + \sec^{2}x*\tan^{2}x\)

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