Assume that has a power-series representation in terms of powers of , verify that it has the form

for all .

First, we recall the triple angle identity for the sine,

Since we know the expansion for is

we then have

In the last two steps, we moved the sum from to infinity to to infinity since the term was 0, and in the final step we reindexed the sum. This was the requested identity.

There is an error in your solution when you divide the second series by 3, Therefore,instead of writing it should be .

Awesome, thanks! Fixed now.