How do you derive the Sum-to-product identities using the Product-to-sum identites?

Question

Any sites about deriving trigonometry identities would be helpful.
Thanks

Answer 1

To quote the wikipedia:

"Replace x by (x + y) / 2 and y by (x – y) / 2 in the product-to-sum formulae."

Answer 2

There are solutions for arbitrarily large N, that's trivial. But I think the case N=6 is interesting. I haven't been able to find any nontrivial solution. I have a small observation however: The equality is a² + b² + c² + d² + e² + f² = abcdef. Note that a² mod 3 is either 0 or 1, so if all of them are not divisible by 3 then a² + b² + c² + d² + e² + f² ≡ 1 + 1 + 1 + 1 + 1 + 1 ≡ 0 which is divisible by 3, so abcdef is divisible by 3, contradiction. ⇒ WLOG a is divisible by 3. If no other variable is divisible by 3, then a² + b² + c² + d² + e² + f² ≡ 0 + 1 + 1 + 1 + 1 + 1 ≡ 2, so abcdef is not divisible by 3, contradiction ⇒ WLOG b is divisible by 3. If no other variable is divisible by 3, then a² + b² + c² + d² + e² + f² ≡ 0 + 0 + 1 + 1 + 1 + 1 ≡ 1, so abcdef is not divisible by 3, contradiction ⇒ WLOG c is divisible by 3. So we may assume that a, b, c are divisible by 3. It would have been amazing if the equality had only the trivial solution.