Showing an identity of two uniform distributed random variables by using characteristic functions and the inversion formula
The CF of $X$ is $\varphi_X(t) = \frac{e^{ita}-e^{-ita}}{2ita} = \frac{\sin(at)}{at}$. Similarly $\varphi_Y(t) = \frac{\sin(bt)}{bt}$.
Note that $\varphi_X \varphi_Y = \varphi_{X+Y}$ when $X$ and $Y$ are independent. By the inversion formula applied to $X+Y$, I think the remaining step is to compute the density* of $X+Y$ at $0$ , which I believe you can show to be $\frac{2 \min\{a,b\}}{4ab}$.
${}^*$e.g., using the convolution formula, or by working with the CDFs by reasoning about areas in the rectangle $[-a,a] \times [-b, b]$