[Math] Evaluate $\sum_{r=0}^n \binom{n}{r}\sin rx \cos (n-r)x$

Question

Evaluate

$$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$

I tried to use binomial identities, but since there are trigonometric terms, I don't have the idea how to approach it.

Can anyone please help?

Answer 1

Hint:

Since $${n\choose r}={n\choose {n-r}}\qquad \text{and}\qquad \sin(a+b)=\sin a\cos b+\sin b\cos a$$ we have $${n\choose r}\sin rx\cdot\cos (n-r)x+{n\choose {n-r}}\sin(n-r)x\cdot \cos rx={n\choose r}\sin nx$$