use a half - angle identity to find the exact value.\ncos 22.5°\ncos 22.5°=□\n(simplify your answer…

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}}). [ \begin{align*} \cos22.5^{\circ}&=\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{2+\sqrt{2}}{4}}\ &=\frac{\sqrt{2+\sqrt{2}}}{2} \end{align*} ]

Answer:

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