How can i prove the following Equality? involving these infinite products

Question

$$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right)= \frac{\cosh(\frac{\pi}{2}\sqrt3)}{3\pi} $$

This can be found here (http://mathworld.wolfram.com/InfiniteProduct.html) Line 22

It is known that $$\cosh(z) = \prod_{n=1}^{\infty} \left(1+\frac{4z^2}{(2n-1)^2\pi^2}\right) $$

via Weierstass Factorization Theorem

So my question is : if i allow z to be $\frac{\pi}{2}\sqrt3$

how can i prove that $$ \frac{1}{3\pi} \prod_{n=1}^{\infty}\left(1+\frac{3}{(2n-1)^2}\right)= \prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right) $$

without going into some really crazy math. Is it even possible? I've been at it all night. Any help would be much appreciated. Thank you!

Answer 1

Your link states

(D. W. Cantrell, pers. comm., Apr. 18, 2006). The first few explicit cases are ..

and mentions your equation as second example. I would study that section to understand the approach. Equation (23) might hold the connection to the hyperbolic cosine expression.