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Chapter 5: Problem 25

Rewrite \(\cos 4 x\) in terms of \(\cos x\).

Short Answer

Expert verified
Therefore, the expression for \( \cos 4x \) in terms of \( \cos x \) is \( 8\cos^4 x - 8\cos^2 x + 1 \).

Step by step solution

01

Apply the Double Angle Identity to \( \cos 4x \)

The first step is to use the double-angle identity for the cosine function to express \( \cos 4x \) as \( 2\cos^2 2x - 1 \)
02

Apply the Double Angle Identity Again

To write the expression in terms of \( \cos x \), you need to apply the double-angle identity once again on \( \cos 2x = 2\cos^2 x - 1 \) to replace \( \cos 2x \) in the first step, resulting in \( 2[2\cos^2 x - 1]^2 - 1 \)
03

Simplify the Expression

Expand and simplify that equation will result in \( 8\cos^4 x - 8\cos^2 x + 1 \)

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