Prove that for all $(n,t,x) \in \mathbb{N}^{*}\times \mathbb{R}\times]0,\pi[$ : $\sin(x) = \left( \sum_{k=1}^{n}t^{k-1}\sin kx \right)\left( 1-2t\cos x+t^{2} \right)+t^{n}\left( \sin((n+1)x)-t\sin(nx) \right)$
Hint : Calculate $\sum_{k=1}^n t^k e^{ikx}$
I have genuinely no idea on how to prove this equality. By working out with the given hint, I find:$$\sum_{k=1}^n t^k e^{ikx}=te^{ix} \times \frac{1-t^ne^{inx}}{1-te^{ix}}$$But this leads me to nowhere. I would gladly appreciate some help.