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

Here is Why You Need An Adapter

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

Azores

In Azores, outlets of type B, C and F are in use.

Type B

Type C

Type F

Cook Islands

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

Type I