<s>[INST] Product of repeated cosec. 
$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$

I realize that there must be some sort of trick in this.
$$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$
I noticed that: $\sin(90 + x) = \cos(x)$ hence,
$$\sin(89) = \cos(-1) = \cos(359)$$
$$\sin(1) = \cos(-89) = \cos(271)$$
$$\cdots$$
$$P \cdot P = \frac{\cos^2(-1)\cos^2(-3)....}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$
But that doesnt help?
 [/INST] I'm using a non-standard notation inspired by various programming languages in this evaluation because I think it's a bit easier to follow than the traditional big pi product notation. Hopefully, it's self-explanatory. Also, all angles are given in degrees.
This derivation uses the following identities:
$$ 
\cos x = \sin(90 - x) \\
\sin(180 - x) = \sin x \\
\sin 2x = 2 \sin x \cdot \cos x \\
\text{and hence} \\
\cos x = \frac{\sin 2x}{2 \sin x}\\
$$
$$\begin{align}
\text{Let } P & = prod(\csc^2 x: x = \text{1 to 89 by 2}) \\
1 / P & = prod(\sin x: x = \text{1 to 89 by 2})^2 \\
 & = prod(\sin x: x = \text{1 to 89 by 2}) \cdot prod(\cos x: x = \text{1 to 89 by 2}) \\
 & = prod(1/2 \sin 2x: x = \text{1 to 89 by 2}) \\
 & = 2^{-45} prod(\sin x: x = \text{2 to 178 by 4}) \\
 & = 2^{-45} prod(\sin x: x = \text{2 to 86 by 4}) \cdot \sin 90 \cdot prod(\sin x: x = \text{94 to 178 by 4})
\end{align}$$
$$
\text{But } prod(\sin x: x = \text{94 to 178 by 4}) = prod(\sin x: x = \text{2 to 86 by 4}) \\
\text{Let } Q = prod(\sin x: x = \text{2 to 86 by 4}), \text{so } P = \frac{2^{45}}{Q^2}
$$
$$\begin{align}
Q & = prod(cos x: x = \text{4 to 88 by 4}) \\
  & = \frac{prod(sin 2x: x = \text{4 to 88 by 4})}{prod(2 sin x: x = \text{4 to 88 by 4})} \\ 
  & = 2^{-22}\frac{prod(sin x: x = \text{8 to 176 by 8})}{prod(sin x: x = \text{4 to 88 by 4})} \\ 
  & = 2^{-22}\frac{prod(sin x: x = \text{8 to 88 by 8}) \cdot prod(sin x: x = \text{96 to 176 by 8})}
  {prod(sin x: x = \text{4 to 84 by 8}) \cdot prod(sin x: x = \text{8 to 88 by 8})} \\
\end{align}$$
$\text{But } prod(\sin x: x = \text{96 to 176 by 8}) = prod(\sin x: x = \text{4 to 84 by 8})$
$\text{Hence } Q = 2^{-22} \text{ and } P = 2^{45} / (2^{-22})^2 = 2^{89}$
</s>