You seem to be missing an important '+' sign, and didn't use parentheses as needed, but I think what you mean is this:

To show this is true start by cancelling out the square root sign and 2A^2 from both sides, leaving:

Now substitute 2*alpha for theta, and use the double angle formula for cosine:

All that remains is some minor rearrangement and resubstitute theta for alpha. Also note that because of the square root signs this is true only for positive values of cos(theta/2): -pi <= theta <= +pi.