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

Here is Why You Need An Adapter

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

Cook Islands

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

Type I

Falkland Islands

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

Type G