Question

Find the exact value using a half angle identity: cos 7 pi/8

Answer 1

Answer:

$\cos \frac{7\pi}{8}=-\frac{1}{2}\sqrt[]{2+\sqrt[]{2}}$

Explanation:

Given

$\cos \frac{7\pi}{8}$

Using the identity:

$\begin{gathered} \cos (\frac{7\pi}{2\times4})=\pm\sqrt[]{\frac{1+\cos\frac{7\pi}{4}}{2}} \\ \\ =-\sqrt[]{\frac{1+\frac{1}{\sqrt[]{2}}}{2}} \\ \\ =-\sqrt[]{\frac{2+\sqrt[]{2}}{4}} \\ \\ =-\frac{1}{2}\sqrt[]{2+\sqrt[]{2}} \end{gathered}$