You Do Need power plug adapter while traveling from Falkland Islands to Cook Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cook Islands is I and in Falkland Islands is G. This is how they look like.

Falkland Islands

In Falkland Islands, outlets of type G are in use.

Type G

Cook Islands

In Cook Islands, outlets of type I are in use.

Type I