Answer
$$\pm\sqrt{\frac{1-\cos5A}{1+\cos5A}}=\tan\frac{5A}{2}$$
Work Step by Step
$$\pm\sqrt{\frac{1-\cos 5A}{1+\cos 5A}}$$
Here we notice at the start of the expression, there is the sign $\pm$, which means we do not have to decide whether to take positive or negative square root.
From the half-angle identity for tangents:
$$\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=\tan\frac{\theta}{2}$$
We can apply the identity to the given expression with $\theta=5A$.
$$\pm\sqrt{\frac{1-\cos5A}{1+\cos5A}}=\tan\frac{5A}{2}$$
