Question

use a half - angle identity to find the exact value.
cos 157.5°
cos 157.5° = □
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Identify the half - angle identity

The half - angle identity for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. Here, $\alpha = 315^{\circ}$ since $\frac{315^{\circ}}{2}=157.5^{\circ}$. Also, $157.5^{\circ}$ is in the second quadrant where cosine is negative.

Step2: Find the value of $\cos\alpha$

We know that $\cos315^{\circ}=\cos(360^{\circ}-45^{\circ})=\cos45^{\circ}=\frac{\sqrt{2}}{2}$.

Step3: Substitute into the half - angle formula

Substitute $\alpha = 315^{\circ}$ into the formula $\cos157.5^{\circ}=-\sqrt{\frac{1+\cos315^{\circ}}{2}}$.

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Answer:

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