Yes you can!
All plastic bottles with a recycling symbol are recyclable regardless of what they contained. This includes oils, solvents,cleaners and cosmetics as well as food items.
To rinse or not to rinse? That is the question. If you live in an urban or sub urban areas with sewers and waste water treatment rinsing is fine. In rural areas you want to minimize detergents in waste water so don’t rinse. Everything does get cleaned before recycling so this is sort of a non issue.