This is from Small's introductory problem book in functional equations which I have been toying with. This question has proved a bit challenging. I was wondering if someone clever could share a hint on which direction this is headed (It's recreational so I'd rather not see a full solution if possible).
I have tried using the angle sum/product trigonometric identities to produce a specific solution but with little luck. I also tried some exponential trial solutions to identify something along that road but as my intuition suggested it appeared not to be the right fit. I am wondering is there is a way to arrive at the conclusion without producing a specific solution. Any thoughts are appreciated!