Answered

Review the following derivation of the tangent double angle identity. The steps are not listed in the correct order.

\begin{tabular}{|c|c|}
\hline
Step 1 & [tex]$\tan (2x) = \frac{\frac{2 \sin (x)}{\cos (x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}}$[/tex] \\
\hline
Step 2 & [tex]$\tan (2x) = \frac{2 \tan (x)}{1 - \tan^2(x)}$[/tex] \\
\hline
Step 3 & [tex]$\tan (2x) = \frac{2 \sin (x) \cos (x)}{\cos^2(x) - \sin^2(x)}$[/tex] \\
\hline
\end{tabular}

What is the correct order of the steps used to derive the identity?

A. [tex]$4, 3, 5, 1, 2$[/tex]

B. [tex]$4, 5, 3, 1, 2$[/tex]

C. [tex]$3, 4, 5, 1, 2$[/tex]

D. [tex]$3, 4, 1, 5, 2$[/tex]



Answer :

GinnyAnswer
Let's review the derivation of the double-angle identity for tangent and see how to sequence the steps correctly. The goal is to derive the identity \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\).

Here are the given steps:

1. \(\tan(2x) = \frac{\frac{2 \sin(x)}{\cos(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}}\)
2. \(\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}\)
3. \(\tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2(x) - \sin^2(x)}\)

Now let's determine the correct order:

Step 3: Start with the double-angle formulas for sine and cosine.
[tex]\[ \tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2(x) - \sin^2(x)} \][/tex]

Step 4: Simplify the equation using trigonometric identities.
[tex]\[ \tan(2x) = \frac{\frac{2 \sin(x)}{\cos(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}} \][/tex]

Step 1: Substitute \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and work through the algebra.
[tex]\[ \tan(2x) = \frac{\frac{2 \sin(x)}{\cos(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}} \][/tex]

Step 5: Simplify the equation further.
[tex]\[ \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \][/tex]

Step 2: Arrive at the final double-angle identity for tangent.
[tex]\[ \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \][/tex]

So, the correct order of the steps used to derive the identity is:
[tex]\[3, 4, 1, 5, 2\][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{3, 4, 1, 5, 2} \][/tex]

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