In mathematics, we often see statements such as $\sin ^2x + \cos^2 x = 1$.
Notice that this is not a sentence, as the variable $x$ is free. But we all agree that this statement is true, given the usual interpretations of the symbols. How can we square this with the claim that sentences are the formulas that can be either true or false?
On
A formula that is always true does not have to be a sentence.
We have e.g. formula $x=x$. True in every model and looked at as logical axiom.
Dealing with sine and cosine we are busy in some model $\mathfrak A$.
There we can easily interpret function symbols in such a way that for distinct terms $t$ and $s$ we have: $$\mathfrak A\vDash t=s[\alpha]$$ for every $\mathfrak A$-assignment $\alpha$.
On
It follows from the definition of satisfaction in a structure:
$$\mathcal{M} \models \phi \\\Longleftrightarrow \text{ for all assignment functions } v: \mathcal{M}, v \models \phi$$
i.e., a formula is true in a structure if and only if it is true under all variable assignments in that structure.
The definition of satisfaction thus renders open formulas as implicitly universally quantified, and we have
$$\mathcal{M} \models \phi \text{ iff } \mathcal{M} \models \forall x_1 \ldots \forall x_n \phi \text{ (where } x_1, \ldots, x_n \text{ are the variables occurring free in } \phi \text{)}$$
that is, a formula is true in a structure exactly when its universal closure is.
So
$$\sin ^2x + \cos^2 x = 1$$
is a distinct formula from but has the same structure-level truth conditions as
$$\forall x (\sin ^2x + \cos^2 x = 1)$$
In practice, a statement such as
$$ \sin^2 x + \cos^2 x = 1 $$
serves as a convenient abbreviation for
$$ \forall x \, \sin^2 x + \cos^2 x = 1. $$
In other words, free variables like $x$ are governed by implicit universal quantifiers. Most of the time, writing out all quantifiers increases clutter and hurts readability, so it is avoided.