use a half - angle formula to find the exact value of \\( \\cos \\frac { 7 \\pi } { 8 } \\).\n\\( \\cos…

Explanation:

Step1: Recall the half - angle formula

The half - angle formula for cosine is (\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}). For (\cos\frac{7\pi}{8}), we have (\alpha=\frac{7\pi}{4}), and (\frac{\alpha}{2}=\frac{7\pi}{8}). Since (\frac{7\pi}{8}) is in the second quadrant ((\frac{\pi}{2}<\frac{7\pi}{8}<\pi)), (\cos\frac{7\pi}{8}<0).

Step2: Find the value of (\cos\alpha)

We know that (\cos\frac{7\pi}{4}=\cos(2\pi-\frac{\pi}{4})=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2})

Step3: Substitute into the half - angle formula

Substitute (\alpha = \frac{7\pi}{4}) into (\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}) (negative because of the quadrant). [ \begin{align*} \cos\frac{7\pi}{8}&=-\sqrt{\frac{1+\frac{\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})