I don't understand them at all.
The problem is to evaluate tan(22.5) using exact values.
How should I do this?
$\displaystyle \tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$
Actually is $\displaystyle \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}},$ but the angle is positive, hence our answer is positive, and that's preserves the positive sign.