To find the point of intersection of the two equations \(y = 2 \sin x - \cos x\) and \(y = \cos x\), we need to set them equal to each other:
\[ 2 \sin x - \cos x = \cos x \]
Rearranging gives us:
\[ 2 \sin x - 2 \cos x = 0 \]
Dividing by 2 leads us to:
\[ \sin x - \cos x = 0 \]
This implies:
\[ \sin x = \cos x \]
The solutions to this equation in the interval \([0, \frac{\pi}{2}]\) occur at:
\[ x = \frac{\pi}{4} \]
We can calculate the corresponding \(y\)-value by substituting \(x = \frac{\pi}{4}\) into either original equation. Let's use \(y = \cos x\):
\[ y = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
Thus, the \(y\)-value of the point of intersection is:
\[ \frac{\sqrt{2}}{2} \]
Given the options, the correct response is:
\[ \frac{\sqrt{2}}{2} \]