I have to proof that the solution for the logistic discrete equation: $x_{n+1}=4x_n\left(1-x_n\right)$ with $r=4$, and $x_0 \in (0,1)$ when $n=0$, it's of the form $$x_n=\sin^2\left(\beta\alpha^n\right)$$ and find the values of $\beta, \alpha$. With the following identities: $\cos^2\theta+\sin^2\theta=1$; $\sin2\theta=2\sin\theta\cos\theta$
So far I know that: $$ x_1 = 4x_0(1-x_0) $$ \begin{align*} x_2 & = 4x_1(1-x_1) \\ & = 4(4x_0(1-x_0))(1-(4x_0(1-x_0))) \\ & = 4^2x_0(1-x_0)(1-4x_0+4x_0^2) \\ & = 4^2x_0(1-x_0)(1-2x_0)^2 \end{align*} $$ \vdots $$ and with the identities I have: $\sin^2(2\theta) = 4\sin^2\theta(1-\sin^2\theta)$.
From that I know that $\alpha =2$, and I read in an article that $\beta = \sin^2(x_0)$. But I don't know what's the next step in the proof, does anyone have an idea?