Michael, the first thing I would suggest you to perform is an extensive keyword research for both the pages, the Home and the Second Page, lets call the Home as A and Second as B.
Do a keyword research and as you said, since both of them have the same content, you will get a bunch of similar keywords (excluding the plurals/singulars). The ones with high search volume can be targeted from A which you think are your business keywords as A will have a better SEO value being the domain. The second bunch of keywords with less searches can be targeted from B. Do keep in mind that an good amount of quality links to A will help you make the page more valuable. It is good to optimize B also as the more optimized the pages are, the better chances of getting more value from the SERPs.
Note - You did mention above that the page have same content, so if they are are entirely the same with respect to content, you better not optimize B and instead place a 301 redirect from there to A or if it is not feasible from the user's point, do add a canonical tag on B. Duplicate pages can harm your traffic and rankings, especially when it is the Home Page.
Hope that sums it up!
Cheers,