find the exact value of $\displaystyle cos(\frac{\pi}{8})$

$\displaystyle \cos\frac{\pi}{8} = \cos\left({\frac{1}{2}*\frac{\pi}{4}}\right)$

$\displaystyle \cos\frac{\pi}{8} = \pm\sqrt{\frac{1 - \sin\frac{\pi}{4}}{2}}$

$\displaystyle \cos\frac{\pi}{8} = \pm\sqrt{\frac{1 -\frac{\sqrt{2}}{2}}{2}}$

$\displaystyle \cos\frac{\pi}{8} = \pm\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}$

$\displaystyle \cos\frac{\pi}{8} = \pm\sqrt{\frac{2 - \sqrt{2}}{4}}$

$\displaystyle \cos\frac{\pi}{8} = \pm\frac{\sqrt{2 - \sqrt{2}}}{2}$

therefore: we take the positive one because $\displaystyle \frac{\pi}{8}$ is in the first quadrant and its positive.

this correct ? much appreciated..