The short answer
Instagram lets you add up to five accounts inside the app and switch between them. It works, it is free, and it takes ten seconds. The catch is that all five live in one app install — one set of stored data, one device, one session — and adding them together is itself a statement that they belong to the same person. For a personal account alongside a business account that is completely fine. If the accounts are meant to be independent, you need real separation, which means containers on a jailbroken phone or separate devices.