Expansion of $\sin(a_{1}+a_{2}+....+a_{n})$?

Question

We know this formula:

$$\sin(a+b)=\sin a\cdot\cos b+\sin b\cdot\cos a$$

So how to find the of the expansion of this

$$\sin(a_{1}+a_{2}+\cdots+a_{n})=\,?$$

Answer 1

$\sin(a+b+c) = \sin((a+b)+c) = \sin(a+b)\cos(c) + \sin(c)\cos(a+b)$. Now use the sum formulae for $\sin$ and $\cos$ again. Try induction after a few specific cases.

Also wolfram alpha will happily expand these for you...