General formula for $\mathrm{sin}({A}_{1}+{A}_{2}+\cdots +{A}_{n})$

I was wondering if there could be a general formula for this. Somehow, I managed to get (after some very small experimentation, so I'm not sure of what I'm going to say) that

$\mathrm{sin}({A}_{1}+{A}_{2}+\cdots +{A}_{n})=\mathrm{cos}{A}_{1}\mathrm{cos}{A}_{1}\cdots \mathrm{cos}{A}_{n}(\sum _{i=1}^{n}\mathrm{tan}{A}_{i}-\sum _{1\le i<j<k\le n}(\mathrm{tan}{A}_{i}\mathrm{tan}{A}_{j}\mathrm{tan}{A}_{k}))$

And I cannot prove it. Can anyone help me prove or disprove it?

I was wondering if there could be a general formula for this. Somehow, I managed to get (after some very small experimentation, so I'm not sure of what I'm going to say) that

$\mathrm{sin}({A}_{1}+{A}_{2}+\cdots +{A}_{n})=\mathrm{cos}{A}_{1}\mathrm{cos}{A}_{1}\cdots \mathrm{cos}{A}_{n}(\sum _{i=1}^{n}\mathrm{tan}{A}_{i}-\sum _{1\le i<j<k\le n}(\mathrm{tan}{A}_{i}\mathrm{tan}{A}_{j}\mathrm{tan}{A}_{k}))$

And I cannot prove it. Can anyone help me prove or disprove it?