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