How do I apply the product (capital pi) symbol here?

That is not the product symbol. It is the incomplete elliptic integral of the third kind (A non-elementary function).

It is defined as:

$$\Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{\left(1-m t^2\right)\left(1-t^2\right) }}$$

You will probably not be able to simplify your result further.

This is not a product: it's an elliptic integral (of the third kind, according to Legendre's scheme), defined by $$ \Pi(n;\varphi \mid m) = \int_0^{\sin{\varphi}} \frac{dt}{(1-nt^2)\sqrt{(1-t^2)(1-mt^2)}}. $$ More information about elliptic integrals may be found on Wikipedia, Mathworld, the DLMF pages, or the book Modern Analysis by Whittaker and Watson (published in the 1920s, now freely available online), Chapter 22. Be aware that the notation varies by source: some use $k^2$ instead of $m$, or something else instead of $\sin{\varphi}$.