Can anyone think of a way to input the following infinite sum into Maxima for computation?

\(\displaystyle \sum_{n=0}^{\infty }2^{n-1}c_n^2\)

(The tricky part...)

where \(\displaystyle c_0=\sin\theta\)

and \(\displaystyle \sin\theta_{n+1}=\frac{1-\cos\theta_n}{1+\cos\theta_n}\)