Answer
The required solution is $2\cos x\cos \frac{x}{2}$.
Work Step by Step
One of the sum-to-product formula is $\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$. so, in this question, according to the above-mentioned formula, the value of $\alpha $ is $\frac{3x}{2}$ and the value of $\beta $ is $\frac{x}{2}$.
Thus, the expression can be evaluated as provided below:
$\begin{align}
& \cos \frac{3x}{2}+\cos \frac{x}{2}=2\cos \frac{\frac{3x}{2}+\frac{x}{2}}{2}\cos \frac{\frac{3x}{2}-\frac{x}{2}}{2} \\
& =2\cos \frac{\frac{4x}{2}}{2}\cos \frac{\frac{2x}{2}}{2} \\
& =2\cos \frac{4x}{4}\cos \frac{2x}{4} \\
& =2\cos x\cos \frac{x}{2}
\end{align}$
Hence, the provided expression can be written as $2\cos x\cos \frac{x}{2}$. So, it is not possible to find the exact value.
