Chapter 5.6, Problem 18E

The half-angle identity

$\tan \frac{A}{2}=\pm \sqrt{\frac{1-\cos A}{1+\cos A}}$ can be used to find $\tan {22.5}^{\circ }=\sqrt{3-2\sqrt{2}},$ and the half-angle identity

$\tan \frac{A}{2}=\frac{\sin A}{1+\cos A}$ can be used to find $\tan {22.5}^{\circ }=\sqrt{2}-1.$

Show that these answers are the same, without using a calculator. (Hint. If a > 0 and b > 0 and a² = b², then a = b.)