Question

use a half - angle formula to find the exact value of \\( \cos 67.5 ^ { \circ } \\).
\\( \cos 67.5 ^ { \circ } = \square \\)

Explanation:

Step1: Identify the half - angle formula

The half - angle formula for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. Since $67.5^{\circ}=\frac{135^{\circ}}{2}$ and $67.5^{\circ}$ is in the first quadrant (where cosine is positive), we use the positive form of the formula. Here $\alpha = 135^{\circ}$.

Step2: Find the value of $\cos135^{\circ}$

We know that $\cos135^{\circ}=\cos(180^{\circ}-45^{\circ})=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}$.

Step3: Substitute into the half - angle formula

Substitute $\cos\alpha=-\frac{\sqrt{2}}{2}$ into $\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}$.

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Answer:

$\frac{\sqrt{2 - \sqrt{2}}}{2}$