#### Answer

The required value of the ${{\cos }^{2}}\alpha $ is $\frac{1+\cos \,2\alpha }{2}$.

#### Work Step by Step

In order to find the value of ${{\cos }^{2}}\alpha $, the power reducing formula is used that is as shown below:
$\cos 2\alpha =2\,{{\cos }^{2}}\alpha -1$
The aforementioned formula is proved in the earlier problems.
Solve the aforementioned formula on the right side for the ${{\cos }^{2}}\alpha $:
$\begin{align}
& 2{{\cos }^{2}}\alpha =1+\cos 2\alpha \\
& {{\cos }^{2}}\alpha =\frac{1+\cos 2\alpha }{2}
\end{align}$
Hence, the required value of the ${{\cos }^{2}}\alpha $ is $\frac{1+\cos \,2\alpha }{2}$.