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Chapter 5: Problem 28

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{4} 2 x$$

Short Answer

Expert verified
The expression \(\sin ^{4} 2 x\) can be rewritten in terms of the first power of the cosine as \(\frac{1}{4}(1 - 2\cos 4x + \frac{1 + \cos 8x}{2})\).

Step by step solution

01

Apply Power-Reducing Formula First Time

Rewrite \(\sin ^{4} 2 x\) as \(\sin ^{2}(2 x)\) squared. Then apply the power-reducing formula to \(\sin^{2}(2 x)\), which yields \((\frac{1 - \cos 4x}{2})^2\).
02

Expand the Power and Simplify the Result

We then need to square \(\frac{1 - \cos 4x}{2}\) to obtain \(\frac{1}{4}(1 - 2\cos 4x + \cos^2 4x)\)
03

Apply Power-Reducing Formula Second Time

Apply the power-reducing formula to \(\cos^2 4x\). Replace \(\cos^2 4x\) with \(\frac{(1 + \cos 8x)}{2}\). Then substitute it into the expression from the previous step to get \(\frac{1}{4}(1 - 2\cos 4x + \frac{1 + \cos 8x}{2})\).

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