#### Answer

$$\pm\sqrt{\frac{1+\cos20\alpha}{2}}=\cos10\alpha$$

#### Work Step by Step

$$\pm\sqrt{\frac{1+\cos20\alpha}{2}}$$
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 cosines:
$$\pm\sqrt{\frac{1+\cos A}{2}}=\cos\frac{A}{2}$$
We can apply the identity to the given expression with $A=20\alpha$.
$$\pm\sqrt{\frac{1+\cos20\alpha}{2}}=\cos\frac{20\alpha}{2}$$
$$\pm\sqrt{\frac{1+\cos20\alpha}{2}}=\cos10\alpha$$