Polynomials with $f(\sin x) = f(\cos x)$
Solution 1:
Yup, I just arrived at that.
Make a substitution $t^2=u$, then we have $f(u)$ can be any polynomial that is symmetric around $u=\frac{1}{2}$.
One example could be $a(u)=(u-\frac{1}{2})^2$ which corresponds to $f(t)=(t^2-\frac{1}{2})^2$.
Another example of a non-polynomial function would be $f(t)=|t^2-\frac{1}{2}|$.
Any polynomial that is a linear combination of even powers of $(t^2-\frac{1}{2})$ will work.