A

Let \( \vec{a} \) and \( \vec{b} \) be the two non-collinear unit vectors. The vectors can be expressed in terms of their magnitudes and angles. Because they are unit vectors, we have \( |\vec{a}| = |\vec{b}| = 1 \). The angle between them is given, and we can also express dot product as \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = \cos \theta \).
Next, since we are required to find the result of \( |\vec{a} + \vec{b}| \), we use the following equation:
\( |\vec{a} + \vec{b}| = \sqrt{ |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} } = \sqrt{ 1 + 1 + 2 \cos(\theta) } = \sqrt{2 + 2 \cos(\theta)}.
Simplifying gives us \( |\vec{a} + \vec{b}| = \sqrt{2(1 + \cos(\theta))} = 2 \cos(\frac{\theta}{2}) \) using the identity \( 1 + \cos \theta = 2 \cos^2(\frac{\theta}{2}) \).
Thus, the answer falls within the provided options; confirming A is the correct response.