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Chapter 5: Problem 43

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\cos ^{4} x$$

Short Answer

Expert verified
\( \cos^{4}x = \frac{1}{4} + \frac{1}{2}\cos(2x) + \frac{1}{4}\cos^2(2x)\)

Step by step solution

01

Power-reducing formula

The power-reducing formula we are going to use is \((\cos^{2}x) = (1 + \cos(2x))/2\). This formula expresses \(\cos^2x\) in terms of \(\cos2x\), reducing the power of the cosine from 2 to 1.
02

Express \(\cos^{4}x\) as \((\cos^{2}x)^2\)

The first step is to turn \(\cos^{4}x\) into \((\cos^{2}x)^2\), because it's now in a format that allows us to apply the power-reducing formula. This doesn't change the value of the expression, it just rearranges it into a form where it matches the left side of the power-reducing formula.
03

Apply the power-reducing formula

We can now apply the power-reducing formula to the modified equation: \((\cos^{2}x)^2 = ((1 + \cos(2x))/2)^2\).
04

Simplify the expression

Finally, simplify the expression to its final form. This involves expanding the squared binomial and simplifying the result, leading to \((\cos^{4}x) = \frac{1}{4} + \frac{1}{2}\cos(2x) + \frac{1}{4}\cos^2(2x)\).

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