<s>[INST] 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.
 [/INST] We have to calculate $$\prod^{45}_{k=1}\sin^2((2k-1)^\circ)$$ 
Because $\sin(180-\theta)=\sin(\theta)$
Now Let $$P=\prod^{45}_{k=1}\sin((2k-1)^\circ)$$
Then $$\prod^{45}_{k=1}\sin(2k^\circ)\cdot P=\prod^{45}_{k=1}\sin((2k-1)^\circ)\cdot \prod^{45}_{k=1}\sin(2k^\circ)$$
So $$\prod^{45}_{k=1}\sin(2k^\circ)\cdot P=\frac{1}{2^{44}}\cdot \frac{1}{\sqrt{2}}\cdot \prod^{45}_{k=1}\sin(2k^\circ)$$
So we get $$P=\frac{1}{2^{\frac{89}{2}}}$$
So $$\prod^{45}_{k=1}\sin^2((2k-1)^\circ)=\frac{1}{2^{89}}$$
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