Is there a known general duplication formula for function $\sin$ and $\cos$ and $\tan$? I mean give an arbitrary family of real numbers $(x_i)_{1\le i\le n}$ do we know formula for $$\sin\left(\sum_{i=1}^{n}x_i\right),~~~\cos\left(\sum_{i=1}^{n}x_i\right),~~~~and ~~~\tan\left(\sum_{i=1}^{n}x_i\right)$$
The case $n=2$ is pretty well know but using induction for $n=3$ is getting messy already.
Now some attempt of generalization for $$\sin\left(nx\right)~~~~\text{and} ~~~~\cos\left(nx\right).$$ can be found here.