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

Here is Why You Need An Adapter

The outlet and plug type used in Cook Islands is I and in Cayman Islands are A and B. This is how they look like.

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B

Cook Islands

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

Type I