What is the value of $\sin 1 ^\circ \sin3^\circ\sin5^\circ \sin 7^\circ \sin 9^\circ \cdots \sin 179^\circ $? 
What is the value of
  $\sin 1 ^\circ \sin3^\circ\sin5^\circ \sin 7^\circ \sin 9^\circ \cdots \sin 179^\circ $ ?

The question is indeed intriguing. We could start by condensing it using $\sin \theta = \sin (180-\theta)$, This reduces the problem as the products till $89^\circ$.  But that doesn't help proceed.  
Thanks in advanced.

As mentioned in a comment by Hans Lundmark in this question, we have
$$
\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)
$$
The product we want is
$$
\prod_{k=0}^{89} \sin\left(\frac{(2k+1)\pi}{180} \right)
=
\prod_{k=0}^{90-1} \sin\left(\frac{\pi}{180} + \frac{k\pi}{90} \right)
=
\frac{\sin\left(90\frac{\pi}{180}\right)}{2^{90-1}}
=
\frac{\sin\left(\frac{\pi}{2}\right)}{2^{89}}
=
\frac{1}{2^{89}}
$$
