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.