Review the proof of de Moivre's theorem (not in order).

\begin{tabular}{|c|c|}
\hline
& Proof of de Moivre's Theorem \\
\hline
& [tex]$[\cos (\theta) + i \sin (\theta)]^{k+1}$[/tex] \\
\hline
A & [tex]$=[\cos (\theta) + i \sin (\theta)]^k \cdot [\cos (\theta) + i \sin (\theta)]^1$[/tex] \\
\hline
B & [tex]$=\cos (k \theta + \theta) + i \sin (k \theta + \theta)$[/tex] \\
\hline
C & [tex]$=\cos (k \theta) \cos (\theta) - \sin (k \theta) \sin (\theta) + i[\sin (k \theta) \cos (\theta) + \cos (k \theta) \sin (\theta)]$[/tex] \\
\hline
D & [tex]$=[\cos (k \theta) + i \sin (k \theta)] \cdot [\cos (\theta) + i \sin (\theta)]$[/tex] \\
\hline
E & [tex]$=\cos [(k+1) \theta] + i \sin [(k+1) \theta]$[/tex] \\
\hline
\end{tabular}

Which steps must be switched to put the proof in order?

A. steps B and C
B. steps B and D
C. steps C and D
D. steps C and E



Answer :

To determine which steps must be switched to put the proof of de Moivre's Theorem in order, let's review the steps carefully:

1. We start with the expression:
[tex]\[ [\cos(\theta) + i \sin(\theta)]^{k+1} \][/tex]

2. Using the properties of exponents, we can break this down as follows:
[tex]\[ [\cos(\theta) + i \sin(\theta)]^{k+1} = [\cos(\theta) + i \sin(\theta)]^k \cdot [\cos(\theta) + i \sin(\theta)] \][/tex]
This is Step A.

3. Next, we apply the distributive property (expanding the product of two complex numbers):
[tex]\[ [\cos(\theta) + i \sin(\theta)]^k \cdot [\cos(\theta) + i \sin(\theta)] = [\cos(k\theta) + i \sin(k\theta)] \cdot [\cos(\theta) + i \sin(\theta)] \][/tex]
This is Step D.

4. We then express it using the angle addition formulas for cosine and sine:
[tex]\[ [\cos(k\theta) + i \sin(k\theta)] \cdot [\cos(\theta) + i \sin(\theta)] = \cos(k\theta) \cos(\theta) - \sin(k\theta) \sin(\theta) + i[\sin(k\theta) \cos(\theta) + \cos(k\theta) \sin(\theta)] \][/tex]
This is Step C.

5. Combining terms, we get:
[tex]\[ \cos((k+1)\theta) + i \sin((k+1)\theta) \][/tex]
This is Step E.

6. Since we know the result of angle addition directly:
[tex]\[ = \cos(k\theta + \theta) + i \sin(k\theta + \theta) \][/tex]
This is Step B.

So, in order for the sequence to be correct, we need to switch Step B and Step D. Specifically:

- Step A remains the same.
- Step D should come right after Step A.
- Step C follows Step D.
- Step B should come before Step C.
- Step E is the final step.

Therefore, the correct answer is:

Steps B and D.