I was given a very interesting sum of series by a friend. The problem is to prove that if $ x \in (0, 2\pi) $ , then
$$ x + \sin x + \sin(x + \sin x) + \sin(x + \sin x + \sin(x + \sin x)) + ... = \pi $$
The $n^{th}$ term is related to the sum of all the previous $n-1$ terms as
$$ T_n = \sin ( S_{n-1} ) $$
I tried doing something using this relation, which didn't yield anything significant.
I also tried using Euler's form of a complex number to somehow convert it into a more manageable series, which also failed miserably.
Any hints or approaches would be appreciated.
The problem is supposed to be at the high school level.