Question

use a double - angle formula to find the exact value of the given expression.
1 - 2\sin^{2}22.5^{circ}
1 - 2\sin^{2}22.5^{circ}=\square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressi

Explanation:

Step1: Recall the double - angle formula

The double - angle formula for cosine is \(\cos(2\alpha)=1 - 2\sin^{2}\alpha\).

Step2: Substitute \(\alpha = 22.5^{\circ}\)

Let \(\alpha=22.5^{\circ}\), then \(1 - 2\sin^{2}22.5^{\circ}=\cos(2\times22.5^{\circ})\).

Step3: Simplify the angle

\(2\times22.5^{\circ}=45^{\circ}\), so \(\cos(2\times22.5^{\circ})=\cos45^{\circ}\).

Step4: Find the value of \(\cos45^{\circ}\)

We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\).

Answer:

\(\frac{\sqrt{2}}{2}\)