What are the differences between equality, equations and identities?

Question

What are the differences between the followings:

1. Identity $$ \sin^2(\alpha) + \cos^2(\alpha) = 1 $$

2. Equation $$ 4x = 16 $$

3. Equality - $x,y$ are mathematical objects. $$ x = y $$

All of the three examples use the same equality sign but they have a different meaning. How do I differentiate between the three, and what does each one of the examples mean?

Edit: I understand the difference between an equation and an identitiy. I want to understand the difference between an equation and identity and an equality of two mathematical objects. I want to know how to ifferentiate between the three because they all use the same equal sign to mean different things.

We all agree that: $x$ and $y$ are matrices and $x=y$ is different from $4x=16$. But why? How do you know that?

Answer 1

It's not the "$=$" that's changing meanings, but rather the (context around the) variables. The term "identity" generally indicates that the equality in question has a universal quantifier around it, while a general equality is just asserted to hold for a particular value (or values) of the variable(s) in question.

Here's a bit more detail:

In the first example, there is an implicit universal quantifier: when we refer to the "identity" $\sin^2(x)+\cos^2(x)=1$, we're really referring to the somewhat-more-detailed statement $$\mbox{For every $x$, we have }\sin^2(x)+\cos^2(x)=1.$$ (Even this isn't fully detailed since we should say what sort of thing $x$ can be - e.g. can $x$ be a complex number here? - but since this isn't the main issue I'm sliding past it.) By contrast, when we're solving an equation like "$4x=16$," the idea is that some particular $x$ satisfies this property and we're trying to figure out what it is.

That said, note the way that solving an equation like this plays with the universal quantifier: when we turn $4x=16$ into $x=4$, we can think of this as proving that the statement $$\mbox{For every $x$, we have }[4x=16\iff x=4]$$ is true. This has the same form as the identity above, but with a more complicated "matrix" (this is an old-timey term for "the bit inside the quantifier(s)").

We usually suppress the "For every" bit for ease of writing, but this makes things much trickier for those (like the OP, and myself way back when) who try to figure out what's "really going on" under the surface.