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Chapter 5: Problem 62

Verify the identity. $$\sec ^{4} x \tan ^{2} x=\left(\tan ^{2} x+\tan ^{4} x\right) \sec ^{2} x$$

Short Answer

Expert verified
The final result \( (\tan^2 x + \tan^4 x)\sec^2 x \) verifies the given identity.

Step by step solution

01

Express sec^4 x in terms of sec^2 x.

We can rewrite \( \sec ^{4} x \) as \( (\sec ^{2} x)^{2} \). This gives us \( (\sec ^{2} x)^{2} \tan ^{2} x \)
02

Use the trigonometric identity sec^2 x = 1 + tan^2 x

We can substitute \( \sec ^{2} x \) with \( 1+ \tan ^{2} x \) in the above expression. This gives us \( (1+ \tan ^{2} x)^{2} \tan ^{2} x \)
03

Expand the square

On expanding \( (1+ \tan ^{2} x)^{2} \) we get \( 1+ 2\tan ^{2} x + \tan ^{4} x \). Thus the expression becomes \( (1+ 2\tan ^{2} x + \tan ^{4} x)\tan ^{2} x \)
04

Distribute \( \tan^2 x \)

Multiply the \( \tan^2 x \) into the parentheses, giving \( \tan^2 x + 2\tan^4 x + \tan^6 x \).
05

Reorganize terms

Reorder the expression to \( \tan^2 x + \tan^4 x + \tan^4 x \).
06

Factor out a sec^2 x

Factor out a \( \sec^2 x = 1+ \tan^2 x \) from the expression, resulting in \( (\tan^2 x + \tan^4 x)\sec^2 x \). This verifies the given identity.

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