Is phase shifting all the harmonics in $f(t)$ by π the same as multiplying $f(t)$ by -1?

Question

Given $sin(\omega t + π) = −sin(\omega t)$

and

$cos(\omega t + π) = −cos(\omega t)$

see
https://en.wikipedia.org/wiki/List_of_trigonometric_identities

Is phase shifting each of the harmonics in a function, f(t), by π the same as multiplying f(t) by -1?

Assume $f(t)$ can be decomposed into harmonics with no constant term.

I think the answer must be yes, but I can not find a reference, identity or proof--would appreciate one of these.