Question 131
Express the following in closed form, as a function of $x$:
$\sin^{2}(x) + \sin^{2}(2x) \cos^{2}(x) + \sin^{2}(4x) \cos^{2}(2x) \cos^{2}(x) + \cdots + \sin^{2}\left(2^{2010} x\right) \cos^{2}\left(2^{2009} x\right) \cdots \cos^{2}(2x) \cos^{2}(x)$.
$\sin^{2}(x) + \sin^{2}(2x) \cos^{2}(x) + \sin^{2}(4x) \cos^{2}(2x) \cos^{2}(x) + \cdots + \sin^{2}\left(2^{2010} x\right) \cos^{2}\left(2^{2009} x\right) \cdots \cos^{2}(2x) \cos^{2}(x)$.