Answered

Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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]

Sagot :

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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.