review the steps in the derivation of the tangent sum identity. the steps are not in order. which list shows…

Explanation:

Step1: Start with tangent definition

We know that $\tan(A + B)=\frac{\sin(A + B)}{\cos(A + B)}$, so we start with step I: $\frac{\sin(x + y)}{\cos(x + y)}$.

Step2: Apply sum - of - angles formulas

Using the sum - of - angles formulas $\sin(x + y)=\sin(x)\cos(y)+\cos(x)\sin(y)$ and $\cos(x + y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, we get $\frac{\sin(x)\cos(y)+\cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)}$. Then, dividing each term in the numerator and denominator by $\cos(x)\cos(y)$ gives step II: $\frac{\frac{\sin(x)\cos(y)}{\cos(x)\cos(y)}+\frac{\cos(x)\sin(y)}{\cos(x)\cos(y)}}{\frac{\cos(x)\cos(y)}{\cos(x)\cos(y)}-\frac{\sin(x)\sin(y)}{\cos(x)\cos(y)}}$.

Step3: Simplify using tangent definition

Since $\tan(x)=\frac{\sin(x)}{\cos(x)}$ and $\tan(y)=\frac{\sin(y)}{\cos(y)}$, the expression simplifies to step III: $\frac{\tan(x)+\tan(y)}{1 - \tan(x)\tan(y)}$. And step IV is the intermediate step of dividing by $\cos(x)\cos(y)$. The correct order is I→IV→II→III.

Answer:

I→IV→II→III