Question

use a half - angle identity to find the exact value. cos 67.5° cos 67.5° = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the half - angle identity

The half - angle identity 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 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 $\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}$. Then $\cos67.5^{\circ}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{2}}{4}}=\frac{\sqrt{2-\sqrt{2}}}{2}$.

Answer:

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