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The only reason this is not true, is if you believe the title is not descriptive enough to enduce someone to click. If not then ad one word to the end. Something like Whatever Widget World or Selection.
You are already doing what 98% of people do not do: concentrate on a specific keyword phrase per page, put the keyword phrase exclusively as your title. And presumably duplicating same effort with H tags and content with a density of 5-20%. This is the easiest recipe in the book and the best (albeit simplified for the purpose of posting).
Meta tag content (in this case we are talking title tag) that does not contain your keyword target phrase should be used only to enduce clicks- it is not directly related to SEO, as it is extraneous to the target, and in fact convolutes the target. As far as keyword density is concerned, one should concern themselves with the target, not what is extraneous to the target. In writing SEO content however, it will become quickly apparent that you will need to MIX up the manner in which you use your target. For EXAMPLE: Start as many sentences off as possible with the target, "Whatever Widgets is pleased to.....' "Whatever widgets offers...... "Whatever widgets product 1".... etc etc.
This recipe works - period and the key is consistency. 1 page = 1 unique target keyword phrase. From title tag to headers 1,2 and 3, to content, to anchor text, consistency and density of efforts is a winning proposition.