Answer
$$\pm\sqrt{\frac{1-\cos\frac{3\theta}{5}}{2}}=\sin\frac{3\theta}{10}$$
Work Step by Step
$$\pm\sqrt{\frac{1-\cos\frac{3\theta}{5}}{2}}$$
From the half-angle identity for sine:
$$\pm\sqrt{\frac{1-\cos A}{2}}=\sin\frac{A}{2}$$
We can apply the identity to the given expression with $A=\frac{3\theta}{5}$.
$$\pm\sqrt{\frac{1-\cos\frac{3\theta}{5}}{2}}=\sin\frac{\frac{3\theta}{5}}{2}$$
$$\pm\sqrt{\frac{1-\cos\frac{3\theta}{5}}{2}}=\sin\frac{3\theta}{10}$$
