Use an identity to write the expression as a single number.

2 cosine squared left parenthesis 22.5 degrees right parenthesis minus 12cos2(22.5°)−1

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Part 1

A.StartFraction StartRoot 2 EndRoot Over 4 EndFraction

StartFraction StartRoot 2 EndRoot Over 4 EndFraction24

B.StartRoot 3 EndRoot

StartRoot 3 EndRoot3

C.StartFraction StartRoot 3 EndRoot Over 3 EndFraction

StartFraction StartRoot 3 EndRoot Over 3 EndFraction33

D.StartFraction StartRoot 2 EndRoot Over 2 EndFraction

StartFraction StartRoot 2 EndRoot Over 2 EndFraction

To simplify the expression \( 2\cos^2(22.5^\circ) - 1 \), we can use a trigonometric identity. Here are the steps:

### Step 1: Use the Double-Angle Identity for Cosine

The expression \( 2\cos^2(x) - 1 \) can be simplified using the double-angle identity for cosine:
\[
2\cos^2(x) - 1 = \cos(2x).
\]

### Step 2: Apply the Identity

For \( x = 22.5^\circ \), we substitute into the identity:
\[
2\cos^2(22.5^\circ) - 1 = \cos(2 \times 22.5^\circ) = \cos(45^\circ).
\]

### Step 3: Find the Value of \( \cos(45^\circ) \)

We know that:
\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}.
\]

So, the expression \( 2\cos^2(22.5^\circ) - 1 \) simplifies to \( \frac{\sqrt{2}}{2} \).

### Answer

The correct answer is:
**D.** \( \frac{\sqrt{2}}{2} \).

Would you like further clarification or details on any of these steps?

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Here are five related questions to expand your understanding:

1. How would you simplify \( 2\sin^2(x) - 1 \) using an identity?
2. What is the value of \( \cos(30^\circ) \) in exact form?
3. How can we use identities to simplify expressions involving \( \sin^2(x) + \cos^2(x) \)?
4. Why is \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \)?
5. Can this type of identity be used for angles not in degrees, such as radians?

**Tip:** Memorizing common trigonometric identities can make simplifying expressions much quicker and easier.

Use an identity to write the expression as a single number: 2cos^2(22.5°) - 1

Learn how to simplify the expression 2cos^2(22.5°) - 1 using the double-angle identity for cosine. This problem covers key trigonometric identities and the value of cos(45°), leading to the simplified result of √2 / 2.

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