Why is the infinite product of this quotient of $\sin$'s equal to $\left(\frac{3}{\pi}\right)^{2}$[SOLVED]

I was intrigued by this answer the other day, but perhaps lack a little bit of the necessary background to understand a certain step.

Namely, the fact that $$\prod_{n=1}^{+\infty} \left(\frac{\sin\left(\frac{\pi}{6\cdot 2^{n-1}}\right)}{2\sin\left(\frac{\pi}{6\cdot 2^n}\right)}\right)^2 = \left(\frac{3}{\pi}\right)^{2}$$ is used, and I cannot make sense of this at all. Jack D'Aurizio comments that "the product of squares is the square of the product" but I'm not entirely sure what this means. Can anyone explain it to me a little more in-depth? I'm not sure what to look up to understand this foreign concept.

Edit: With a hint from Simon S I got the answer, so I have to thank him very much. I'm not sure what to do with my question now.

• "the product of squares is the square of the product" just means (ab)^2 = a^2 b^2 – Simpson17866 Nov 10 '15 at 16:40
• Edit the title with SOLVED in bracket – Kushal Bhuyan Nov 10 '15 at 17:07
• @KprimeX Thanks, I will. – WeiYing Nov 10 '15 at 17:08
• @WeiYing If I might give a suggestion, it would be helpful if you would answer your own question by explaining how you solved your question in the Answer box. – Hrodelbert Nov 10 '15 at 17:13
• @Hrodelbert Oh alright, for sure. I won't be free for an hour or two but I'll come back and post my findings then, thanks for the suggestion. – WeiYing Nov 10 '15 at 17:15