Example 5 Evaluate $(1) \sin 1^{\circ} \sin 3^{\circ} \sin 5^{\circ} \cdots \sin 87^{\circ} \sin 89^{\circ}$
(2) $\sin 1^{\circ} \sin 2^{\circ} \sin 3^{\circ} \cdots \sin 88^{\circ} \sin 89^{\circ}$.
(1) Let $x=\sin 1^{\circ} \sin 3^{\circ} \sin 5^{\circ} \cdots \sin 87^{\circ} \sin 89^{\circ}$,
$y=\sin 2^{\circ} \sin 4^{\circ} \sin 6^{\circ} \cdots \sin 88^{\circ} \sin 90^{\circ}$,
then
$$
\begin{aligned}
x \cdot y & =\sin 1^{\circ} \sin 2^{\circ} \sin 3^{\circ} \cdots \sin 87^{\circ} \sin 88^{\circ} \sin 89^{\circ} \sin 90^{\circ} \\
& =\left(\sin 1^{\circ} \cdot \sin 89^{\circ}\right) \cdot\left(\sin 2^{\circ} \cdot \sin 88^{\circ}\right) \cdots\left(\sin 44^{\circ} \cdot \sin 46^{\circ}\right) \cdot \sin 45^{\circ} \\
& =\left(\sin 1^{\circ} \cos 1^{\circ}\right) \cdot\left(\sin 2^{\circ} \cos 2^{\circ}\right) \cdots\left(\sin 44^{\circ} \cos 44^{\circ}\right) \cdot \sin 45^{\circ} \\
& =\frac{1}{2^{44}} \sin 2^{\circ} \sin 4^{\circ} \sin 6^{\circ} \cdots \sin 88^{\circ} \cdot \frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2^{45}} \cdot y .
\end{aligned}
$$

Since $y \neq 0$, we have $x=\frac{\sqrt{2}}{2^{45}}$, i.e., $\sin 1^{\circ} \sin 3^{\circ} \sin 5^{\circ} \cdots \sin 89^{\circ}=\frac{\sqrt{2}}{2^{45}}$.
(2) Continuously use $\sin 3 \theta=4 \sin \theta \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)$ to reduce the angles.
$$
\begin{array}{r}
\sin 1^{\circ} \sin 2^{\circ} \sin 3^{\circ} \cdots \sin 89^{\circ}=\left(\sin 1^{\circ} \sin 59^{\circ} \sin 61^{\circ}\right) \cdot\left(\sin 2^{\circ} \sin 58^{\circ} \sin 62^{\circ}\right) \cdots\left(\sin 29^{\circ} \cdot \sin 31^{\circ}\right. \\
\left.\sin 89^{\circ}\right) \cdot \sin 30^{\circ} \sin 60^{\circ}=\frac{\sqrt{3}}{4^{30}} \sin 3^{\circ} \sin 6^{\circ} \sin 9^{\circ} \cdots \sin 87^{\circ}=\frac{\sqrt{3}}{4^{30}}\left(\sin 3^{\circ} \sin 57^{\circ} \sin 63^{\circ}\right) \cdot\left(\sin 6^{\circ} \sin 54^{\circ}\right. \\
\left.\sin 66^{\circ}\right) \cdots\left(\sin 27^{\circ} \sin 33^{\circ} \sin 87^{\circ}\right) \cdot \sin 30^{\circ} \cdot \sin 60^{\circ}=\frac{3}{4^{40}} \sin 9^{\circ} \sin 18^{\circ} \sin 27^{\circ} \cdots \sin 72^{\circ} \cdot \sin 81^{\circ},
\end{array}
$$

and $\sin 9^{\circ} \sin 18^{\circ} \sin 27^{\circ} \cdots \sin 72^{\circ} \sin 81^{\circ}=\left(\sin 9^{\circ} \sin 81^{\circ}\right) \cdot\left(\sin 18^{\circ} \sin 72^{\circ}\right) \cdot\left(\sin 27^{\circ} \cdot\right.$ $\left.\sin 63^{\circ}\right) \cdot\left(\sin 36^{\circ} \sin 54^{\circ}\right) \cdot \sin 45^{\circ}$
$=\left(\sin 9^{\circ} \cos 9^{\circ}\right) \cdot\left(\sin 18^{\circ} \cos 18^{\circ}\right) \cdot\left(\sin 27^{\circ} \cos 27^{\circ}\right) \cdot\left(\sin 36^{\circ} \cos 36^{\circ}\right) \cdot \sin 45^{\circ}$
$=\left(\frac{1}{4}\right)^{2} \cdot \frac{\sqrt{2}}{2} \sin 18^{\circ} \sin 36^{\circ} \sin 54^{\circ} \sin 72^{\circ}=\left(\frac{1}{4}\right)^{2} \cdot \frac{\sqrt{2}}{2} \cdot\left(\sin 18^{\circ} \sin 54^{\circ}\right) \cdot\left(\sin 36^{\circ} \sin 72^{\circ}\right)$
$=\left(\frac{1}{4}\right)^{2} \cdot \frac{\sqrt{2}}{2} \cdot \frac{1}{4} \cdot \frac{\sqrt{5}}{4}=\left(\frac{1}{4}\right)^{5} \cdot 2 \sqrt{10}$ (using the conclusion from the previous example).
Thus, $\sin 1^{\circ} \sin 2^{\circ} \sin 3^{\circ} \cdots \sin 88^{\circ} \sin 89^{\circ}=\left(\frac{1}{4}\right)^{45} \cdot 6 \sqrt{10}=\frac{3 \sqrt{10}}{2^{89}}$.
Note: From the solution processes of (1) and (2), we can see that: $\sin 2^{\circ} \sin 4^{\circ} \sin 6^{\circ} \cdots \sin 88^{\circ} \sin 90^{\circ}=\frac{3 \sqrt{5}}{2^{44}}$.