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Chapter 5: Problem 74

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

Short Answer

Expert verified
The simplified expression using the half-angle formula is \( \cos 2x \).

Step by step solution

01

Identify the half-angle formula

The given expression is \(\sqrt{\frac{1+\cos4x}{2}}\). It's already in the form of the half-angle formula, \(\sqrt{\frac{1+\cos(x)}{2}}\), for the cosine function. Thus, it represents the cosine of half of the angle in the cosine function of the given expression.
02

Apply the half-angle formula

Now, implement the half-angle formula. This expression is actually the cosine of half the angle in the cosine function in the expression. So, if you let \(4x\) be the \(x\) in the half-angle formula, you get \(2x\) when you take half of it because \(\frac{4x}{2} = 2x\). Therefore, \(\sqrt{\frac{1+\cos 4x}{2}} = \cos 2x\).

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