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Chapter 5: Problem 31

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$

Short Answer

Expert verified
The simplified expression in terms of the first power of cosine is \((1 - (cos^{2}(4x)))/4\).

Step by step solution

01

Identify and Apply Power-Reducing Formulas

The power-reducing formulas can be applied to \(sin^{2}(2x)\) and \(cos^{2}(2x)\) separately. Apply the formulas: \(sin^{2}(2x) = (1 - cos(4x))/2\) and \(cos^{2}(2x) = (1 + cos(4x))/2\).
02

Substitution

Substitute \(sin^{2}(2x)\) and \(cos^{2}(2x)\) with their respective power-reducing formulas in the original expression \(sin^{2}(2x) cos^{2}(2x)\). The expression becomes: \((1 - cos(4x))/2 * (1 + cos(4x))/2\).
03

Simplify the Expression

Now, simplify the expression. As per the formula \((a-b)(a+b)=a^{2}-b^{2}\), simplify the expression. The resultant simplified expression is: \((1 - (cos^{2}(4x)))/4\).

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