Don't use your calculators ladies and gentlemen! Again.

Solution:

Formula: $\cos{x} = \sqrt{ \frac{\cos{2x} + 1}{2} }$ .

$cos{135°} = -\frac{\sqrt{2}}{2}$ . Apply this formula. Get $\cos{67.5°}$ . Apply it again. Get $\cos{33.75°}$ . The first one is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ and the latter one is $\frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}}{2}$ . And the latter one is the solution.