<s>[INST] Continued product in $\sin$ series Find the value of the product $$(\sin 1°)(\sin 3°)(\sin 5°)\ldots(\sin 89°)$$ I tried multiplying and dividing by $2$ and then combining and then converting into cosine, but doesn't work out. 
 [/INST] Notice for any integer $N > 0$, we have
$$\begin{align}
z^{2N} + 1 
&= \prod_{k=-N}^{N-1} \left( z - e^{\frac{2k+1}{2N}\pi i} \right)
= \prod_{k=0}^{N-1}\left(z - e^{\frac{2k+1}{2N}\pi i}\right)\left(z - e^{-\frac{2k+1}{2N}\pi i}\right)\\
&= \prod_{k=0}^{N-1}\left[ z^2+1 - 2z\cos\left(\frac{2k+1}{2N}\pi\right)\right]
\end{align}
$$
Set $z = 1$, we find
$$2 
= 2^N \prod_{k=0}^{N-1} \left[1 - \cos\left(\frac{2k+1}{2N}\pi\right)\right]
= 2^{2N} \prod_{k=0}^{N-1} \sin^2\left(\frac{2k+1}{4N}\pi\right)\\
$$
Since all the $\sin(\cdots)$ involved are positive, this leads to
$$\prod_{k=0}^{N-1} \sin\left(\frac{2k+1}{4N}\pi\right) = 2^{\frac12 - N}$$
Substitute $N = 45$, we get
$$\sin(1^\circ)\sin(3^\circ)\cdots\sin(89^\circ)
= \prod_{k=0}^{44}\sin\left(\frac{2k+1}{180}\pi\right) = 
2^{-89/2}$$
</s>