Question

use a half - angle identity to find the exact value.
cos 22.5°
cos 22.5°=□
(simplify your answer, including any radicals. use integers or fractions for any numbers in the ex

Explanation:

Step1: Identify the half - angle identity

The half - angle identity for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). Since \(22.5^{\circ}=\frac{45^{\circ}}{2}\) and \(22.5^{\circ}\) is in the first quadrant (where cosine is positive), we use the positive form of the identity. Here \(\alpha = 45^{\circ}\).

Step2: Substitute \(\alpha = 45^{\circ}\) into the identity

We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\). Substitute into \(\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}\), so \(\cos22.5^{\circ}=\sqrt{\frac{1 + \cos45^{\circ}}{2}}\).

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Answer:

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