Compatibility summary
You will need a travel adapter in Cook Islands; plug types used there differ from Philippines.
Plug comparison
Philippines plug types: A B C
Cook Islands plug types: I
There is no common plug type between Philippines (A B C) and Cook Islands (I). You need a travel adapter.
Voltage comparison
Philippines: 220V, 60Hz
Cook Islands: 240V, 50Hz
Voltage difference is within a range many devices tolerate; check your appliance label.