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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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I don't understand what you mean by "hold identically in that..." – Qiaochu Yuan Aug 23 '12 at 8:03
you can do as $sin(2*x)=sin(x+x)$ and use formula for $sin(a+b)$ – dato datuashvili Aug 23 '12 at 8:04
@Qiaochu Yuan "in so much that" – poirot Aug 23 '12 at 8:05
An identity is precisely an equation which holds identically. If you know what an identity is, then you know what it means for something to hold identically. "Identically" is just the adverbial form of "identity" in this case. – Rahul Aug 23 '12 at 8:06
In the case you're interested in, it means that there is always an equality for all admissible values of $x$... – J. M. Aug 23 '12 at 8:13
up vote 4 down vote accepted

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.

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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.

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