Answer
The positive square root should be selected.
$$\tan225^\circ=\sqrt{\frac{1-\cos450^\circ}{1+\cos450^\circ}}$$
Work Step by Step
$$\tan225^\circ=\pm\sqrt{\frac{1-\cos450^\circ}{1+\cos450^\circ}}$$
Whether the positive or negative square root should be selected depends the sign of $\tan225^\circ$.
$225^\circ$ lies in quadrant III. In quadrant III, both $\cos\theta\lt0$ and $\sin\theta\lt0$. Therefore, $\tan\theta=\frac{\sin\theta}{\cos\theta}\gt0$.
So, $\tan225^\circ\gt0$.
Thus, the positive square root should be selected. In other words,
$$\tan225^\circ=\sqrt{\frac{1-\cos450^\circ}{1+\cos450^\circ}}$$
