trigonometry - General Sine and Cosine formula for sum of a finite number of angles

I was wondering is there is a general formula for $\sin(x_1+x_2+x_3+...+x_n)$ as well as for the cosine function. I know that $\sin(x_1+x_2)=\sin(x_1)\cos(x_2)+\cos(x_1)\sin(x_2)$ and $\cos(x_1+x_2)=\cos(x_1)\cos(x_2)-\sin(x_1)\sin(x_2)$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern. Any suggestions and hints (not answers) will be appreciated.