Products of trig functions and the Thue–Morse sequence

Question

I was studying transformations of finite products of trig functions into sums, and empirically observed that the following curious identity appears to hold for all non-negative integer $m$: $$\prod_{n=0}^m \sin\left(\frac z{2^n}\right)=\frac1{2^m}\sum _{n=0}^{2^m-1} (-1)^{\large t_n} \sin\left(\frac{\pi\,m}2+\frac{2\,n+1}{2^m}\,z\right),\tag{$\diamond$}$$ where $t_n$ is the Thue–Morse sequence.$^{[1]}$$\!^{[2]}$$\!^{[3]}$ How can we prove this identity?