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]
