Find conditional expectation given symmetric r.v

Question

How to find $E[\sin(X)|\cos(X)]$, if X - a symmetric random variable?

My attempt: I tried to prove that $\sin(X)$ is $\sigma(\cos(X))$-measurable. Then it can be argued that $E[\sin(X)|\cos(X)]=\sin(X)$. But in this case, symmetry is not used.

I am completely sure that this is the wrong answer.

Answer 1

By symmetry $$E(\sin (X) |\cos (X))$$ $$=E(\sin (-X) |\cos (-X))=E(-\sin (X) |\cos (X))$$ $$=-E(\sin (X) |\cos (X))$$ so $E(\sin (X) |\cos (X))=0.$