As an example, the following expression $$\sin2x=2\sin x\cos x$$ is a trigonometric identity. Because it is an identity we can replace $x$ by $ax$ and differentiate with respect to $a$ to get $$2x\cos2ax=2x\cos^2ax-2x\sin^2ax.$$ Then, let $a=1$ to obtain another well known identity $$\cos2x = \cos^2x-\sin^2x.$$ But what does it mean to hold "identically" ? Is there a definition ?
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It depends on the context. Usually it means that the LHS and RHS describe functions $f, g$ from some set $X$ to some other set $Y$ and the claim is that they are precisely the same function, which is equivalent to saying that $f(x) = g(x)$ for all $x \in X$.
Sometimes it doesn't mean this. For example, if we say that $f(x) = g(x)$ identically where $f, g$ are polynomials, it means that all of their coefficients are equal. This is not equivalent to saying that $f(x) = g(x)$ for all $x$ if $f, g$ are polynomials over a finite field. In other words, the notion of equality implicit here is equality in a ring of polynomials.
In this case it means that equality is true regardless of the value of $x$, as opposed to being true of the particular values of $x$ being considered.