How to calculate an integral containing $\delta^{(n)}$?

Question

I've searched and found almost nothing about how to integrate the integral containing $\delta^{(n)}$. I wanted to calculate $$\delta^{(n)}*\mathcal{F}_x(J_\alpha(x))$$ where $J_\alpha$ is the Bessell function type 2. I have calculated the Fourier transform of the $J_\alpha$ function and I wanted to calculate the convolution of one of its parts which is $$\delta^{(n)}*\lgroup\int\limits_0^\pi cos((n+1/2)\tau)(\delta(\omega-sin\tau))d\tau\rgroup$$ I searched in https://en.wikipedia.org/wiki/Dirac_delta_function#Distributional_derivatives and I thought that if we use the point mentioned we obtain again $$\int\limits_0^\pi cos((n+1/2)\tau)(\delta^{(n)}(\omega-sin\tau))d\tau$$ which contains $\delta^{(n)}$. I can't move forward and stuck here. Can anyone help me?