Placing the two sides of any equality into almost any functional or algebraic form as if they are variables of that function

There are several things to be careful of here both in real problems in science and in abstract problems in mathematics. In science the usual warnings about units hold as in the previous example. It may look OK to say $x = y$ where $x$ and $y$ are both in meters and then to form $\sin(x) = \sin(y)$, but the latter has a power series expansion and is dimensional nonsense. In mathematics one has to worry about the domain and range (defined below when we talk about functions). Suppose I have the relation $y = 2 + x^2$ where $x$ is a real dimensionless expression, and I wish to take the $\cos^{-1}$ of both sides. Well, the range of cosine is only $-1$ to $1$, and my function $y$ is clearly strictly larger than 2.