Answer
The required value of the $\cos \frac{y}{2}$ is $\pm \sqrt{\frac{1+\cos \,y}{2}}$ .
Work Step by Step
In order to find the value of $\cos \frac{y}{2}$, the half angle formula is used that is as follows:
${{\cos }^{2}}\alpha =\frac{1+\cos 2\alpha }{2}$
Now replace $\alpha $ with $\frac{y}{2}$
$\begin{align}
& {{\cos }^{2}}\frac{y}{2}=\frac{1+\cos 2\left( \frac{y}{2} \right)}{2} \\
& =\frac{1+\cos \,y}{2} \\
& =\pm \sqrt{\frac{1+\cos \,y}{2}}
\end{align}$
