Step 1: Use a half-angle identity instead of finding $y$ first.
We are given $\cos y = 0$ and asked for $\frac{1}{2}\cos\frac{y}{2}$. Rather than identifying $y$ as $90^{\circ}$ and then halving it, we can go straight to $\cos\frac{y}{2}$ using the half-angle formula
\[ \cos\frac{y}{2} = \pm\sqrt{\frac{1 + \cos y}{2}} \]
Step 2: Substitute the given value of $\cos y$.
Since $\cos y = 0$:
\[ \cos\frac{y}{2} = \sqrt{\frac{1 + 0}{2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]
We take the positive root because $y$ lies between $0^{\circ}$ and $180^{\circ}$ for this type of problem, making $\frac{y}{2}$ an acute angle with a positive cosine.
Step 3: Multiply by $\frac{1}{2}$ as required.
\[ \frac{1}{2}\cos\frac{y}{2} = \frac{1}{2} \times \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]
Final Answer:
The value of $\frac{1}{2}\cos\frac{y}{2}$ is $\frac{1}{2\sqrt{2}}$, which corresponds to option (D).
\[ \boxed{\dfrac{1}{2\sqrt{2}}} \]