\section*{Problem 2 - 071232}

It is the product

\[
\sin 5^{\circ} \sin 15^{\circ} \sin 25^{\circ} \sin 35^{\circ} \sin 45^{\circ} \sin 55^{\circ} \sin 65^{\circ} \sin 75^{\circ} \sin 85^{\circ}
\]

to be transformed into an expression that can be formed from natural numbers solely by applying the operations of addition, subtraction, multiplication, division, and root extraction with natural exponents.

Example: \(\sin 30^{\circ} \sin 60^{\circ}=\frac{1}{4} \sqrt{3}\)
}

I use

\[
\sin x \sin y=\frac{1}{2}(\cos (x-y)-\cos (x+y))
\]

and

\[
\sin x \cos y=\frac{1}{2}(\sin (x-y)+\sin (x+y))
\]

With the double-angle sine function and the known sine values for \(30^{\circ}\) and \(45^{\circ}\), the product can initially be written as \(\frac{1}{64} \sqrt{2} \sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}\). The remaining product of sine values can be simplified to:

\[
\begin{aligned}
\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ}= & \frac{1}{2}\left(\cos 40^{\circ}-\cos 60^{\circ}\right) \sin 70^{\circ}=\frac{1}{4}\left(2 \cos 40^{\circ} \sin 70^{\circ}-\sin 70^{\circ}\right)= \\
& =\frac{1}{4}\left(\sin 30^{\circ}+\sin 110^{\circ}-\sin 70^{\circ}\right)=\frac{1}{8}
\end{aligned}
\]

Thus, the product value is \(\frac{1}{512} \sqrt{2}\).