AEF.ERU: Equations Related to Unity (1)
The relation of quotients equal to unity (1) is given by the simple but important theorem that is a result of other basic field properties of real numbers.
Theorem QU: For any $a,b \in R, b\ne 0$, $a \div b = \frac ab=1$ if and  only if $a=b$.
A simple corollary of Theorem QU: For $f(x)=f_1(x) \div f_2(x)=\frac{f_1(x)}{f_2(x)}, f(x) = 1$ if and  only if $f_1(x)=f_2(x)$.
For an elementary function, $f$ that can be expressed in a form with the final step in its definition being the quotient of two elementary functions, $f_1$ and $f_2$, the corollary entails further that set of the roots of $f$ is the set of roots of the equation, $f_1(x)=f_2(x)$.

A mapping diagram can visualize these results by visually connecting coincident arrows in the mapping diagrams for $f_1$ and $f_2$ to a root of $f$.
Example AEF.QU.1  Visualize the solutions of $f(x)=(x^2-4) \div \sin(x)=\frac{x^2-4}{\sin(x)} = 1$ for $x \in [-4,4]$.