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 ?
What does it mean for an expression to hold "identically"?
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0I don't understand what you mean by "hold identically in that..." – 2012-08-23
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0you can do as $sin(2*x)=sin(x+x)$ and use formula for $sin(a+b)$ – 2012-08-23
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0@Qiaochu Yuan "in so much that" – 2012-08-23
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0I don't understand what that means either. Are you trying to prove the identity? I don't see how what you've written constitutes a proof. – 2012-08-23
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1An 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. – 2012-08-23
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0No, the first part gives an example of something we can do if the equation holds identically. What I am asking is what is the definition of "holds identically", which will hopefully help me understand why it is we can do what we do in the first part. – 2012-08-23
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1In the case you're interested in, it means that there is always an equality for all admissible values of $x$... – 2012-08-23
2 Answers
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.