Evaluate :
`1-2 sin ^(2) 22.5 ^(@)`

Answer

To evaluate the expression \( 1 - 2 \sin^2(22.5^\circ) \), we can use the double angle identity for cosine. The identity states that:

\[ \cos(2a) = 1 - 2 \sin^2(a) \]

### Step-by-Step Solution:

1. **Identify the angle**: In our case, \( a = 22.5^\circ \).

2. **Apply the double angle identity**: Using the identity, we can rewrite the expression:
\[ 1 - 2 \sin^2(22.5^\circ) = \cos(2 \times 22.5^\circ) \]

3. **Calculate the angle**: Now calculate \( 2 \times 22.5^\circ \):
\[ 2 \times 22.5^\circ = 45^\circ \]

4. **Evaluate cosine**: Now we need to find \( \cos(45^\circ) \):
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

5. **Final result**: Therefore, we can conclude that:
\[ 1 - 2 \sin^2(22.5^\circ) = \frac{1}{\sqrt{2}} \]

### Final Answer:
\[ \frac{1}{\sqrt{2}} \]