Don't use your calculator ladies and gentlemen!

First of all, I must say that @Drex Beckman , with his solution for one of my problems forced me to think in this direction. Therefore, I think it is appropriate to put his name in here :)

Solution: There is a formula which states: $cos x = \sqrt{ \frac{cos 2x + 1}{2} }$ . I think it is a common knowledge that $cos 30° = \frac{\sqrt{3}}{2}$ . Apply this formula, to get $cos 15°$ and afterwards to get $cos 7.5°$ . The first one is $\frac{\sqrt{\sqrt{3} + 2}}{2}$ and the latter one is $\frac{\sqrt{\sqrt{\sqrt{3} + 2} + 2}}{2}$ . And the latter one is the solution.