fourier transform - Multiplication property DTFT

I was truing to solve an example of DTFT which is following multiplication property. The problem is

$$a^n \sin(\omega_0 n) u[n]$$ we know that the definition of DTFT is $$X(j \omega) = \sum _ {n=-\infty} ^ {{+\infty}} x[n]e^{-j \omega n}$$ Multiplication in Time domain will be convolution in DTFT. if we take the DTFT of $a^n u[n]$ we have $\frac {1}{1-ae^{-j \omega}}$ and DTFT of $\sin(\omega_0 n) u[n]$ will be $\frac {\pi}{j} \sum _ {l=-\infty} ^ {{+\infty}} \delta(\omega + \omega_0 - 2\pi l) - \delta(\omega - \omega_0 - 2\pi l)$

I have confusion how can I write it in the form of multiplication property.