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

Here is Why You Need An Adapter

The outlet and plug type used in Cook Islands is I and in Faroe Islands are C, E, F and K. This is how they look like.

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K

Cook Islands

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

Type I