Step 1: Information Retrieval.
The provided text explicitly identifies the family of coconut as Arecaceae.
Step 2: Option Exclusion.
- Gramineae: This family encompasses cereals such as wheat, rice, and maize.
- Leguminosae: This family is associated with pulses.
- Anacardiaceae: This family includes plants like mango and cashew.
Step 3: Determination.
Consequently, Arecaceae is established as the family to which coconut belongs.
Final Determination: \[\boxed{\text{Coconut's Family = Arecaceae}}\]