A detailed outline for your letter:
Paragraph 1: Greet your cousin warmly. Something like, Hello, it's been a while. How have you been? would be appropriate. For the sake of persuasion, do not bring up music lessons in your greeting.
Paragraph 2: Without proposing the offered music lessons, write to your cousin about both of your feelings about music. Music lessons cost time and money, but it would be a worthwhile investment, and, after all, you know your cousin has the discipline to nurture this talent.
Paragraph 3: Neutrally explain who is offering music lessons. Include all the terms such as lesson times/dates, costs, who the teacher is, location of lessons, and answers to any questions your cousin might have so that they aren't left wondering anything.
Paragraph 4: Like you did in the greeting, close out nicely. Perhaps ask more questions about their personal life or share about your own. Say you look forward to hearing them play/sing a certain song in the future. Then, sign off.