But then, vascop, that's a different case, because you have fixed a value of x from the start (and never mind the point that maths notation shouldn't be based on people who don't know about it). However, since you didn't need to do that in order to make (cos x)^2 + (sin x)^2 = 1, as a starting point it's an identity rather than an equation.
Having said that, I tend to write it with two lines not three. In most practical cases, it isn't going to matter.
One recent example where it did was the following problem:
Let - 8x + p ≡ 2xq + q^2
In standard mathematical rules, x is the "free variable" in this problem, and p and q are some cofficients with fixed, but unknown, values. Hence, the identity ≡ between the two sides, rather than the equality =, allows us to assume that the statement above is true for any value of x. Hence, what looks like one equation in three unknowns becomes (by, say, choosing any two values for x you like), two equations in two unknowns. As an identity, the statement above implies that q=-4 and p=16; an an equality, you could find a form for p in terms of q and x, but nothing else.