Happy Christmas: Is it Really True?

[George Keeling]

TL;DR

sine and cosine power trees

Is this really true? It resembles the binomial theorem.
I've posted it twice which might be breaking rules. Happy Christmas PF.

 

[TeethWhitener]

https://en.m.wikipedia.org/wiki/Chebyshev_polynomials

 

[George Keeling]

https://en.m.wikipedia.org/wiki/Chebyshev_polynomials

Chebyshev was very good for cosines but not so useful for sines.