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Chapter 5: Problem 42

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

Short Answer

Expert verified
The simplified expression is \(cos(2x)\).

Step by step solution

01

Identify the half-angle formula

The half-angle formula for cosine is \( \cos^2(2x) = (1 + \cos (4x))/2 \). This is the formula that fits the expression provided in the exercise.
02

Apply the half-angle formula

By comparing the given expression with the half-angle formula, it can be seen that the expression inside the square root is similar to the right side of the half-angle formula. Thus, we can rewrite the expression \(\sqrt{(1+\cos 4 x)/2} \) as \( \cos(2x) \).
03

Final expression

After applying the half-angle formula, the final expression is \( \cos(2x) \).

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