Question

what is the exact value of \\( \cos ( \frac { 7 \pi } { 8 } ) \\)?

\\( - \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 } \\)

\\( - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 } \\)

\\( - \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } \\)

\\( - \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } \\)

Explanation:

Step1: Use the half - angle formula

The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\).
We know that \(\frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2}\), and \(\cos\frac{7\pi}{8}<0\) (since \(\frac{7\pi}{8}\) is in the second quadrant where cosine is negative).

Step2: Substitute \(\alpha=\frac{7\pi}{4}\) into the formula

We know that \(\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\).
Substitute into \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}\) (negative because of the quadrant), we get \(\cos\frac{7\pi}{8}=-\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}\).

Step3: Simplify the expression

$$
LATEXBLOCK0
$$

Answer:

D. \(-\frac{\sqrt{2 + \sqrt{2}}}{2}\)