Five Claims on Chebyshev-Like Maps

For functions on $[-1, 1]$, consider the claims:

1. if $f(\cos x) = \cos nx$ with $n$ odd, then $f$ is odd; 2. if $f(\cos x) = \cos nx$ with $n$ even, then $f$ is even; 3. some odd $n$ and odd function $g$ satisfy $g(\sin x) = \sin nx$ for all $x$; 4. some even $n$ and even function $g$ satisfy $g(\sin x) = \sin nx$ for all $x$; 5. some $n$ makes both $f$ (with $f(\cos x) = \cos nx$) and $g$ (with $g(\sin x) = \sin nx$) monotone.

Which claims are true?