use a half - angle identity to find the exact value.\ncos 157.5°\ncos 157.5° = □\n(simplify your answer…

Explanation:

Step1: Identify the half - angle identity

The half - angle identity for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. 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 $\cos157.5^{\circ}=-\sqrt{\frac{1+\cos315^{\circ}}{2}}$. [ \begin{align*} \cos157.5^{\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:

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